In order not to have to worry about size issues, I'm going to answer the following question instead:
For a (small) cardinal number $\kappa$, is the category of small categories with $\kappa$-small 2-colimits 2-cocomplete?
If you take $\kappa$ to be inaccessible, then this will correspond to your question, under a particular choice of foundations. I presume moreover that you mean "2-colimits" in the weak "up-to-equivalence" sense which the nLab uses (which 2-category theorists traditionally call "bicolimits").
The fact that the 2-category Cat of small categories is 2-cocomplete, in this sense, has been well-known to category theorists for decades. It is obvious that Cat is cocomplete as a 1-category (since it is locally finitely presentable), and since it is closed symmetric (cartesian) monoidal, it follows by general enriched category theory that it is cocomplete as a category enriched over itself. In the nLab terminology, it has all strict 2-colimits. We then observe that strict pseudo 2-limits, which are 2-limits that represent cones commuting up to isomorphism but satisfy their universal property up to isomorphism (rather than equivalence), are particular strict 2-limits. Since any strict pseudo 2-limit is also a (weak) 2-limit, Cat is 2-cocomplete.
Now as Zoran pointed out in the comments, there is a 2-monad on Cat whose algebras are categories with $\kappa$-small colimits; let us call this 2-monad $T$. The strict $T$-morphisms are functors which preserve colimits on-the-nose, while the pseudo $T$-morphisms are those which are $\kappa$-cocontinuous in the usual sense (preserve colimits up to isomorphism). Therefore, the question is whether the 2-category $T$-Alg of $T$-algebras and pseudo $T$-morphisms is 2-cocomplete.
The answer is yes: it was proven by Blackwell, Kelly, and Power in the paper "Two-dimensional monad theory" that for any 2-monad with a rank (preserving $\alpha$-filtered colimits for some $\alpha$) on a strictly 2-cocomplete (strict) 2-category, the 2-category $T$-Alg is (weakly) 2-cocomplete. The 2-monad $T$ has a rank (namely, $\kappa$, more or less), so their theorem applies. I believe this all works just as well in the enriched setting.
No a answere but a specification and description of the issue (too long for a comment).
The free cocompletion property of $Ind(\mathcal{C})$ dont use the hypothesis "$\mathcal{C}$ has finite colimits". ANyway $Ind(\mathcal{C})$ is equivalent to the full subcategory $P_1(\mathcal{C})\subset \mathcal{C}^>$ of presheves that are coimits of a small, filtred diagram of representable. In this way the inclusion $i$ corresponds to the yoneda inclusion $h_-: \mathcal{C}\to P_1(\mathcal{C})$ and it preserves limit, and finite colimits.
The inclusion $P_1(\mathcal{C})\subset \mathcal{C}^>$ create (small) filter colimits (if $P=\varinjlim_{i\in I}P_i$ is a filtrant colimit with $O_i\in P_1(P_1(\mathcal{C})$ then combining the comma categories $P_1(\mathcal{C})\downarrow P_i$ we
make a filtrant (small) diagram of $P_1(\mathcal{C})\downarrow P_i$
Is $F: \mathcal{C}\to \mathcal{D}$ where $\mathcal{D}$ as filter colimits we have a (iso)unique extentions $F': P_1(\mathcal{C})\to \mathcal{D}$ with $F'(P):=\varinjlim_{\ (X, x)\in \mathcal{C}\downarrow P} F(X)$, if $P$ has the Ind-representation $(X_i)_{i\in I}$ i.e. $P=\varinjlim_i h_{X_i}$ then the diagram of the $h_{X_i}$ is a final diagram on the comma category $\mathcal{C}\downarrow P$ then $F'(P):=\varinjlim_i F(X_i)$
If $\mathcal{C}$ has finite colimits $P_1(\mathcal{C})\cong Cart(\mathcal{C}^{op}, Set)$ the latter is the the category cartesians presheaves i.e. that maps finite colimits of $\mathcal{C}$ to finite limits in $Set$, of course the embedding $Cart(\mathcal{C}^{op}, Set)\subset \mathcal{C}^>$ create limits .
All above as a dual version for $Proj(\mathcal{C}):=(Ind(\mathcal{C}^{op}))^{op}$, it is a free completion of $\mathcal{C}$, and if $\mathcal{C}$ has finite limits $Proj(\mathcal{C})$ is equivalent to $Cart(\mathcal{C}, Set)^{op}$ (dual to the category of copresheaves that preserving finite limits) it has (small) colimits and we have the embedding $\iota:=(h^-)^{op}: \mathcal{C}\to Cart(\mathcal{C}, Set)^{op} $.
Now the inclusion $h_-: \mathcal{C}\to Cart(\mathcal{C}^{op}, Set)$ and $\iota: \mathcal{C}\to Cart(\mathcal{C}, Set)^{op}$ induce for the universal properties of these two completions the functors $U: Cart(\mathcal{C}, Set)^{op}\to Cart(\mathcal{C}^{op}, Set)$ with $U(Q)=\varprojlim_{\ h^Y\to Q} h_Y$, and
$F: Cart(\mathcal{C}^{op}, Set)\to Cart(\mathcal{C}, Set)^{op}$ with
$F(P)=\varprojlim_{\ h_X\to P} h^X$ (the limit is in $\mathcal{C}^{<}$ the copresheaves category)
we have the natural isomorphisms:
$Cart(\mathcal{C}, Set)^{op}(F(P), Q)=Cart(\mathcal{C}, Set)(Q, F(P))=$
$\mathcal{C}^{<}(\varinjlim_{\ h^Y\to Q}h^Y, \varprojlim_{\ h_X\to P}h^X)=$
$\varprojlim_{\ h_X\to P} \varprojlim_{h^Y\to Q}\mathcal{C}^<(h^Y, h^X)\cong$
$\varprojlim_{\ h_X\to P} \varprojlim_{h^Y\to Q}\mathcal{C}^>(h_X, h_Y)\cong$
$\mathcal{C}^>(\varinjlim_{\ h_X\to P} h_X, \varprojlim_{h^Y\to Q}h_Y) \cong$
$Cart(\mathcal{C}^{op}, Set)(P, U(Q))$
then $U$ is a adjoint to $F$.
Best Answer
Like all fields of mathematics, Category Theory is not immune to foundational questions. Although different foundations have been presented, the most common foundation for Category Theory is within Set Theory: Category Theory starts with a given universe of sets and then develops its theory. The computation of limits, colimits, and whatnot will be affected by the overlying structure of sets. However, this overlying structure of sets is not variable. When a computation depends on certain principles of Set Theory, it must be labeled as such. For example, Grothendieck Universes and Vopenka's Principle are large cardinal axioms which have direct applications in Category Theory.
That said, Category Theory is in a unique position to deal with independence results that arise from Set Theory. Indeed, every forcing construction in Set Theory has an analogue in Category Theory via sheaves over an appropriate site. More precisely, forcing poset can be viewed as a small category and when endowed with the double-negation topology, the Grothendieck topos of sheaves over this site is equivalent to the Boolean-valued model that one obtains in via forcing. Thus independence results from Set Theory are directly visible to Category Theory by doing the computations inside a Boolean Grothendieck topos instead of the topos of sets.