This question is similar to
Conditions such that taking global sections of line bundles commutes with tensor product?
and
Tensor product of invertible sheaves
except that I am concerned with smooth manifolds. If $M$ is a smooth manifold and $V$ and $W$ are (smooth, real) vector bundles on $M$, are there reasonable conditions that ensure that the map $\Gamma(M,V)\otimes_{\mathscr{O}_M(M)}\Gamma(M,W)\rightarrow\Gamma(M,V\otimes W)$ is an isomorphism? Equivalently, are there any reasonable conditions under which taking global sections of finite locally free $\mathscr{O}_M$-modules commutes with tensor product?
I don't really know much smooth manifold theory, and my intuition is mostly derived from algebraic geometry, where the analogous question, as is indicated in the questions I linked to, is not a simple one, even for invertible sheaves. I'd always assumed the situation for smooth manifolds would be the same, but I was just looking at the Wikipedia page on vector-valued differential forms, and it seems (unless I'm misunderstanding which is certainly possible) to claim there that for $V$ an arbitrary vector bundle and $W$ the $p$-th exterior power of the cotangent bundle, $p\geq 1$, the answer to my question is "yes," with the justification being that "$\Gamma(M,-)$ is a monoidal functor." Looking at the definition of a monoidal functor, it seems to me that all this implies is that there is a map from the tensor product of the global sections to the global sections of the tensor product (which is clear from the definitions) and not that this map is an isomorphism.
Best Answer
This is true without any further assumptions on $M,V,W$ and discussed quite thoroughly in chapter 7 of Conlon's wonderful book "Differentiable Manifolds". The proof given in the text is quite elegant so I can't resist recalling the argument.
There is an obvious $C^{\infty}(M)$-bi-linear map $\alpha \colon \Gamma(V) \times \Gamma(W) \rightarrow \Gamma(V \otimes W)$ given by $\alpha(\xi,\eta)_p = \xi_p \otimes \eta_p$ which gives rise to a map $\alpha \colon \Gamma(V) \otimes_{C^{\infty}(M)} \Gamma(W) \rightarrow \Gamma(V \otimes W)$. We want to show that $\alpha$ is an isomorphism.
When $V$ and $W$ are trivial bundles of rank $n$ and $k$ respectively, one can choose $n$ pointwise linearly independent sections $\xi_i \in \Gamma(V)$ and $k$ pointwise linearly independent sections $\eta_j \in \Gamma(W)$ and then $\Gamma(V \otimes W)$ is easily seen to be a free $C^{\infty}(M)$ module with basis $\{ \xi_i \otimes \eta_j \}_{i,j}$ and so $\alpha$ is an isomorphism of $C^{\infty}(M)$ modules.
If $V$ and $W$ are not trivial, we interpret them as a direct summands of some trivial bundles. That is, we find vector bundles $V^{\perp}$ and $W^{\perp}$ such that $V \oplus V^{\perp}$ and $W \oplus W^{\perp}$ are trivial bundles. Then, by looking at the diagram
$$ \require{AMScd} \begin{CD} \Gamma((V \oplus V^{\perp}) \otimes (W \oplus W^{\perp})) @<{\tilde{\alpha}}<< \Gamma(V \oplus V^{\perp}) \otimes_{C^{\infty}(M)} \Gamma(W \oplus W^{\perp}) \\ @AAA @AAA \\ \Gamma(V \otimes W) @<{\alpha}<< \Gamma(V) \otimes_{C^{\infty}(M)} \Gamma(W) \end{CD} $$
we see that $\tilde{\alpha}$ is injective by the previous paragraph, and we can check directly that both vertical arrows (that are defined with the help of the injective bundle maps $v \mapsto (v,0)$ of $V \rightarrow V \oplus V^{\perp}$ and $w \mapsto (w,0)$ of $W \rightarrow W \oplus W^{\perp}$) are injective, showing that $\alpha$ is injective. Similarly, by reversing the direction of the vertical arrows and replacing them with projections, we see that $\alpha$ is also surjective.
The crux of the proof is the result that every bundle $V$ over $M$ can be realized as a subbundle of a trivial bundle $F$ (stated in terms of the module of global sections, this means that $\Gamma(V)$ is a $C^{\infty}(M)$-projective module). This can be shown by constructing an epimorphism $\psi \colon F \rightarrow V$ of vector bundles from a trivial bundle $F$ onto $V$ and using a fiber metric to split $F$ as an inner direct sum $F = \ker(\psi) \oplus \ker(\psi)^{\perp}$ with $\ker(\psi)^{\perp} \cong V$.
The construction of $\psi$ uses partition of unity. When $M$ is compact, one takes a partition of unity $\{\lambda_i\}_{i=1}^n$ subordinate to a cover $\{U_i\}_{i=1}^n$ of $M$ over which $V$ trivializes with generating global sections $\xi_i^j \in \Gamma(U_i,E|_{U_i})$. Then, one can define global sections $\sigma_i^j = \lambda_i \xi_i^j$ by zero extension outside $U_i$. We obtain finitely many sections, and by taking the trivial bundle with the vector space $X = \mathrm{span} \{ \sigma_i^j \} \subseteq \Gamma(V)$ as fiber, and defining $\psi \colon M \times X \rightarrow V$ as $\psi(p,\sum \sigma_i^j) = \sigma_i^j(p)$ we obtain the required map.
Addendum: