Well, for any universe U, the U-small sets satisfy the axioms of ZFC (or whatever your preferred set theory is). Therefore, anything that we can prove in ZFC about the Yoneda embedding into "all" presheaves will also be true about the Yoneda embedding into U-small presheaves. So on that score, the answer would seem to be "no."
There might be other interpretations that would make the answer "yes," since in vanilla ZFC every class is definable, whereas not every U-large set is definable in terms of U-small ones. Thus an informal statement like "for every large thingumbob X, ..." might be true in ZFC, but no longer provable relative to a universe, since the meaning of "large" has changed (unless we change the quantifier to "for every U-small-definable large thingumbob X"). This doesn't contradict the first observation, since such a statement cannot be a single theorem of ZFC, only a meta-theorem, and each instance of the meta-theorem is about a particular definable class and therefore still true about the corresponding U-small-definable U-large set. However, right now I can't think of any interesting or useful properties of the Yoneda embedding that would fall under this heading.
So I think that probably the answer is still "no."
After some research, I think it has not been observed until now. However, all of the bricks needed to make the argument are almost ready.
In paper "Monoidal bicategories and Hopf algebroids" Brain Day and Ross Street defined a notion of convolution in the context of Gray monoids. For a reason that shall become clear later, I am willing to call it "virtual convolution". Here is the definition. Let $\langle A, \delta \colon A \rightarrow A \otimes A, \epsilon \colon A \rightarrow I \rangle$ be a weak comonoid, and $\langle B, \mu \colon B \otimes B \rightarrow B, \eta \colon I \rightarrow B \rangle$ be a weak monoid in a monoidal bi-category with tensor $\otimes$ and unit $I$, then $\langle \hom(A, B), \star, i \rangle$ is a monoidal category by:
\begin{array}{ccc}
f\star g &=& \mu \circ (f \otimes g) \circ \delta \newline
i &=& \eta \circ \epsilon
\end{array}
So the "convolution structure" exists only virtually --- on $\hom$-categories. If the monoidal bi-category admits all right Kan liftings, then such induced monoidal category $\langle \hom(I, B), \star, i \rangle$ for trivial comonoid on $I$ is monoidal (bi)closed by:
$$f \overset{L}\multimap h = \mathit{Rift}_{\mu \circ (f \otimes \mathit{id})}(h)$$
$$f \overset{R}\multimap h = \mathit{Rift}_{\mu \circ (\mathit{id} \otimes f)}(h)$$
Taking for the monoidal bi-category the bi-category of profunctors, we obtain the well-known formula for convolution. However, in the general setting, such induced structure is far weaker than one would wish to have --- for example in the category of profunctors enriched over a monoidal category $\mathbb{V}$ the induced convolution instead of giving a monoidal structure on the category of enriched presheaves:
$$\mathbb{V}^{B^{op}}$$
merely gives a monoidal structure on the underlying category:
$$\hom(I, \mathbb{V}^{B^{op}})$$
Actually, there is a work-around for this issue in the context of enriched categories, as suggested in the paper, but the general weakness of "virtual convolution" is obvious.
The solution is to find a way to "materialize" the convolution. I shall sketch the idea for internal categories. I think all of the following works for split fibrations and split structures, so let me replace the codomain fibration $\mathbb{C}^\rightarrow \rightarrow \mathbb{C}$ from the question by its split version corresponding to the internal "family functor":
$$\mathit{fam}(\mathbb{C}) \colon \mathbb{C}^{op} \rightarrow \mathbf{Cat}$$
Likewise, for a category $A$ internal to $\mathbb{C}$ I shall write:
$$\mathit{fam}(A) \colon \mathbb{C}^{op} \rightarrow \mathbf{Cat}$$
for the functor corresponding to the externalisation of $A$. We want to show that given a promonoidal structure $$\langle A, \mu \colon A \times A \nrightarrow A, \eta \colon 1 \nrightarrow A \rangle$$
there is a corresponding monoidal closed structure on:
$$\mathit{fam}(\mathbb{C})^{\mathit{fam}(A)^{op}}$$
which just means, that each fibre of $\mathit{fam}(\mathbb{C})^{\mathit{fam}(A)^{op}}$ is a monoidal closed category and reindexing functors preserve these monoidal structures. By fibred Yoneda lemma, for $K \in \mathbb{C}$:
$$\mathit{fam}(\mathbb{C})^{\mathit{fam}(A)^{op}}(K) = \mathit{Prof}(K, A)$$
where $K$ is interpreted as a discrete internal category. There is a correspondence:
$$\mathit{Prof}(K, A) \approx \mathit{Prof}(1, K^{op} \times A) = \mathit{Prof}(1, K \times A)$$
where the last equality holds because $K^{op} = K$ for any discrete category $K$. Since $K$ has a trivial promonoidal structure:
$$K \times K \overset{\Delta^*}\nrightarrow K$$
we obtain a "product" promonoidal structure on $K \times A$:
\begin{array}{rcc}
K \times A \times K \times A &\overset{\Delta^* \times \mu}\nrightarrow& K \times A \newline
1 &\overset{\langle !^*, \eta \rangle}\nrightarrow& K \times A
\end{array}
In more details, since $\mathbb{C}$ is cartesian, every object $K \in \mathbb{C}$ carries a unique comonoid structure:
\begin{array}{l}
K \overset{\Delta}\rightarrow K \times K \newline
K \overset{!}\rightarrow 1
\end{array}
which has a promonoidal right adjoint structure $\langle \Delta^\*, !^\* \rangle$ in the (bi)category of internal profunctors. The product of the above two promonoidal structures is given by the usual cartesian product of internal categories (note, it is not a product in the bicategory of internal profunctors) followed by the internal product functor $\mathit{fam}(\mathbb{C}) \times \mathit{fam}(\mathbb{C}) \overset{\mathit{prod}}\rightarrow \mathit{fam}(\mathbb{C})$.
Then by "virtual convolution" there is a monoidal (bi)closed structure on $\mathit{Prof}(1, K^{op} \times A)$. Therefore each fibre $\mathit{fam}(\mathbb{C})^{\mathit{fam}(A)^{op}}(K)$ is a monoidal (bi)closed category. It is easy to check that reindexing functors preserve these structures.
Let me work out the concept of internal Day convolution in case $\mathbb{C} = \mathbf{Set}$ and a promonoidal structure on a small category is monoidal. The split family fibration (or more accurately, the indexed functor corresponding to the family fibration) for a locally small category $A$:
$$\mathit{fam}(A) \colon \mathbf{Set}^{op} \rightarrow \mathbf{Cat}$$
is defined as follows:
\begin{array}{rcl}
\mathit{fam}(A)(K \in \mathbf{Set}) &=& A^K \newline
\mathit{fam}(A)(K \overset{f}\rightarrow L) &=& A^L \overset{(-) \circ f}\rightarrow A^K\newline
\end{array}
where $K, L$ are sets and $K \overset{f}\rightarrow L$ is a function between sets. One may think of category $A^K$ as of the category of $K$-indexed tuples of objects and morphisms from A. Now, given any monoidal structure on a small category $$\langle A, \otimes \colon A \times A \rightarrow A, I \colon 1 \rightarrow A \rangle$$
the usual notion of convolution induces a monoidal structure on $\mathbf{Set}^{A^{op}}$:
$$\langle F, G \rangle \mapsto F \otimes G = \int^{B, C \in A} F(B) \times G(C) \times \hom(-, B \otimes C)$$
The split fibration:
$$\mathit{fam}(\mathbf{Set})^{\mathit{fam}(A)^{op}} \colon \mathbf{Set}^{op} \rightarrow \mathbf{Cat}$$
may be characterised as follows:
\begin{array}{rcl}
\mathit{fam}(\mathbf{Set})^{\mathit{fam}(A)^{op}}(K \in \mathbf{Set}) &=& \mathbf{Set}^{A^{op} \times K} \newline
\mathit{fam}(\mathbf{Set})^{\mathit{fam}(A)^{op}}(K \overset{f}\rightarrow L) &=& \mathbf{Set}^{A^{op} \times L} \overset{(-) \circ (\mathit{id} \times f)}\rightarrow \mathbf{Set}^{A^{op} \times K}\newline
\end{array}
Since $\mathbf{Set}^{A^{op} \times K} \approx (\mathbf{Set}^{A^{op}})^K$ we may think of $\mathbf{Set}^{A^{op} \times K}$ as of $K$-indexed tuples of functors ${A^{op} \rightarrow \mathbf{Set}}$. In fact:
$$\mathit{fam}(\mathbf{Set})^{\mathit{fam}(A)^{op}} \approx \mathit{fam}(\mathbf{Set}^{A^{op}})$$
It is natural then to extend the monoidal structure induced on $\mathbf{Set}^{A^{op}}$ pointwise to $(\mathbf{Set}^{A^{op}})^K$:
$$(F \otimes G)(k) = \int^{B, C \in A} F(k)(B) \times G(k)(C) \times \hom(-, B \otimes C)$$
where $k \in K$.
On the other hand, using the internal formula for convolution, we get (up to a permutation of arguments):
\begin{array}{c}
\int^{B, C \in A, \beta, \gamma \in K} F(B, \beta) \times G(C, \gamma) \times \hom(\Delta(k), \langle \beta, \gamma \rangle) \times \hom(-, B \otimes C) \newline\hline\newline\hline
\int^{B, C \in A, \beta, \gamma \in K} F(B, \beta) \times G(C, \gamma) \times \hom(k, \beta) \times \hom(k, \gamma) \times \hom(-, B \otimes C) \newline\hline\newline\hline
\int^{B, C \in A} F(B, k) \times G(C, k) \times \hom(-, B \otimes C) \newline
\end{array}
where the first equivalence is the definition of diagonal $\Delta$ --- recall that the diagonal $\Delta(k) = \langle k, k \rangle$ is represented by profunctor $\hom(\langle \overset{1}-, \overset{2}-\rangle, \Delta(\overset{3}-))$, which has profunctorial right adjoint $\hom(\Delta(\overset{1}-), \langle \overset{2}-, \overset{3}-\rangle) \approx \hom(\overset{1}-, \overset{2}-) \times \hom(\overset{1}-, \overset{3}-)$ --- and the second one is by "Yoneda reduction" applied twice.
Final remarks:
Seeing the above proof, one may wonder where the assumptions about the category $\mathbb{C}$ from the question were actually used:
local cartesian closedness guaranteed existence of all right Kan liftings in the bi-category of internal profunctors; without this assumption, the induced monoidal structure on $\mathit{fam}(\mathbb{C})^{\mathit{fam}(A)^{op}}$ would be generally non-closed; to see that local cartesian closedness is really crucial here, recall that fibration $\mathit{fam}(\mathbb{C})$ is a cartesian closed fibration iff $\mathbb{C}$ is locally cartesian closed ---- this means that without local cartesian closedness even trivial convolution of the monoidal structure on the terminal category is not closed; moreover, which has not been stated in the answer, local cartesian closedness made it possible to speak about internal Yoneda embedding
finite colimits (coequalisers) allowed us to define compositions of internal profunctors
To really obtain a split monoidal closed structure via convolution without moving through the equivalence between Gray monoids and monoidal bi-categories ("Coherence for Tricategories", Gordon, Power, Street), one has (of course!) to replace the monoidal bi-category of internal profunctors by equivalent Gray monoid consisting of internal categories of presheaves and internally cocontinous functors.
I think that the right setting for the concept of Day convolution is a "Yoneda monoidal bi-triangle" as sketched in this answer.
Best Answer
I don't see the need to try to make vector spaces into categories. I would just say that in each case we have a closed symmetric monoidal category (respectively Vect or Cat), a map f : X ⊗ Y → Z for some objects X, Y, Z (respectively $\langle-,-\rangle$ : V ⊗ V → R and Hom : Cop × C → Set) and we are forming the associated map X → hom(Y,Z) (where hom denotes the internal hom functor). The double dual construction is obtained by setting Y = hom(X,Z) and letting f be the evaluation map; it doesn't depend on anything but X and Z.
That said, there is a great analogy between Vect and Cat, where R and Set play parallel roles: but what corresponds to the construction sending C to Hom(Cop, Set) is the free vector space functor from Set to Vect. The analogy goes something like this. (I am omitting some technical conditions for convenience.)
I am not claiming there is a way to take an object in one column and get a corresponding object in the other column (although under some circumstances that may be possible): rather that it is fruitful to use the left-hand column as a way of thinking about the right-hand column.
See this nLab page for an introduction to these ideas.