Let me preface this by saying that I am a simple man, with simple needs, and I would appreciate
a) no modifications to the question unless strictly necessary (but please let me know if I need to change something to make it sensible/have a reasonable answer),
b) as simple of an answer as is possible.
Suppose that I have an indexed collection of sites $\{\mathcal{C}_i\}_{i\in I}$ (by a site I mean a category with a Grothendieck pre-topology) where $I$ is some index category. Suppose that for every morphism $f\colon i\to j$ in $I$ I have a morphism of topoi
$$(f_\ast,f^\ast)\colon \mathbf{Sh}(\mathcal{C}_i)\to\mathbf{Sh}(\mathcal{C}_j),$$
which is something like a lax $2$-functor (for $i\xrightarrow{f}j\xrightarrow{g}k$ I have chosen equivalences $(g\circ f)^\ast=f^\ast\circ g^\ast$ and $(g\circ f)_\ast= g_\ast\circ f_\ast$ satisfying 'reasonable compatibility conditions'). I can then form the $2$-limit
$$2\text{-lim}\, \mathbf{Sh}(\mathcal{C}_i),$$
with respect to the pullback functors $f^\ast$. My question is whether or not this is naturally $\mathbf{Sh}(\mathcal{C})$ for some reasonable site $\mathcal{C}$. I strongly suspect this is related to the existence of colimits in $\mathbf{ShTopos}$, as in a paper of Moerdijk, but I find that to be too jargony for me. It could, be related to the colimit of sites, but this seems slightly wrong to me.
Any suggestions would be helpful!
Allow me the use of "jargon". Let $\textbf{Topos}$ be the 2-category of sheaf toposes. (A bit more explicitly: the objects are categories satisfying Giraud's axioms and the morphisms $f : \mathcal{X} \to \mathcal{Y}$ are functors $f^* : \mathcal{Y} \to \mathcal{X}$ that preserve colimits and finite limits, and the 2-cells are the natural transformations of such functors.)
Lemma 1. If a diagram $\mathcal{X} : \mathcal{I} \to \textbf{Topos}$ has a colimit, then the underlying category of that diagram is the limit of the diagram $\mathcal{X} : \mathcal{I}^\textrm{op} \to \textbf{Cat}$ of underlying categories.
Proof. Let $\mathcal{O} = [\textbf{FinSet}, \textbf{Set}]$. It can be shown that, for any sheaf topos $\mathcal{X}$, the category of geometric morphisms $\mathcal{X} \to \mathcal{O}$ is equivalent to the underlying category of $\mathcal{X}$ itself, 2-naturally in $\mathcal{X}$. In other words, the "underlying category" 2-functor $\textbf{Topos}^\textrm{op} \to \textbf{Cat}$ is represented by $\mathcal{O}$. But representable functors preserve any limits that exist, so colimits in $\textbf{Topos}$ must be limits in $\textbf{Cat}$. ◼
In the very simplest cases, it is easy to give an explicit site for the colimit. (However, we do have to use here the more general notion of site where we do not require the underlying category to have pullbacks.)
Proposition 2. Coproducts exist in $\textbf{Topos}$.
Proof. The obvious construction works: choose sites for each of the summands, then form the sheaf topos over their disjoint union. ◼
Proposition 3. Tensors exist in $\textbf{Topos}$.
Proof. Again, there is an obvious construction. By lemma 1, if the tensor of the topos $\textbf{Psh} (\mathcal{C})$ by a small category $\mathcal{I}$ exists, then it must be $\textbf{Psh} (\mathcal{I}^\textrm{op} \times \mathcal{C})$, because: $$[\mathcal{I}, \textbf{Psh} (\mathcal{C})] \cong [\mathcal{I} \times \mathcal{C}^\textrm{op}, \textbf{Set}] \cong \textbf{Psh} (\mathcal{I}^\textrm{op} \times \mathcal{C})$$ In general, the tensor of $\textbf{Sh} (\mathcal{C})$ by $\mathcal{I}$ is $\textbf{Sh} (\mathcal{I}^\textrm{op} \times \mathcal{C})$, where the covering sieves of $\mathcal{I}^\textrm{op} \times \mathcal{C}$ are those whose projections to $\mathcal{C}$ are covering sieves in $\mathcal{C}$. ◼
However, it seems to me in general we have to appeal to Giraud's theorem to show that the candidate category is a sheaf topos, rather than finding an explicit site. (In some sense, though, finding a site is as hard as showing that Giraud's axioms are satisfied: after all, the full subcategory spanned by the small generating set in Giraud's axioms is a site of definition for the topos. But in practice we can appeal to black box theorems to get generators, so we cannot so easily extract a site from the proof.)
Proposition 4. Pushouts (in the bicategorical sense) exist in $\textbf{Topos}$.
Proof. Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{X} \to \mathcal{Z}$ be geometric morphisms. Suppose we have a pullback diagram in $\textbf{Cat}$ (in the bicategorical sense): $$\require{AMScd} \begin{CD} \mathcal{W} @>{k^*}>> \mathcal{Y} \\ @V{h^*}VV @VV{f^*}V \\ \mathcal{Z} @>>{g^*}> \mathcal{X} \end{CD}$$ It is straightforward to check that the category $\mathcal{W}$ satisfies the exactness part of Giraud's axioms, and that the functors $h^* : \mathcal{W} \to \mathcal{Z}$ and $k^* : \mathcal{W} \to \mathcal{Z}$ preserve colimits and finite limits. It can also be shown – e.g. using the theory of locally presentable categories – that $\mathcal{W}$ has a small generating set. So $\mathcal{W}$ is also a sheaf topos, and we have geometric morphisms $h : \mathcal{Z} \to \mathcal{W}$ and $k : \mathcal{Z} \to \mathcal{W}$. Then we have to check that $\mathcal{W}$ has the appropriate universal property with respect to geometric morphisms, but this is also straightforward. ◼
Corollary 5. Colimits (in the bicategorical sense) exist in $\textbf{Topos}$. ◼
I suppose I should give an example to illustrate that things really are tricky. Let $\mathbb{Z}\textbf{-Set}$ be the category of sets equipped with a $\mathbb{Z}$-action. Then we have a pullback diagram (in the bicategorical sense) in $\textbf{Cat}$ of the form below, $$\begin{CD} \mathbb{Z}\textbf{-Set} @>>> \textbf{Set} \\ @VVV @VVV \\ \textbf{Set} @>>> \textbf{Set} \times \textbf{Set} \end{CD}$$ where both arrows $\mathbb{Z}\textbf{-Set} \to \textbf{Set}$ are the forgetful functor and both arrows $\textbf{Set} \to \textbf{Set} \times \textbf{Set}$ are the diagonal embedding. These categories are all presheaf toposes and these functors are all inverse image functors of geometric morphisms, and (appealing to lemma 1 and proposition 4) this corresponds to a pushout diagram (in the bicategorical sense) in $\textbf{Topos}$.
But $\textbf{Set}$ is the category of sheaves on a point $\{ * \}$, and $\textbf{Set} \times \textbf{Set}$ is the category of sheaves on two points $\{ a, b \}$, and the diagonal embedding corresponds to the unique continuous map $2 \to 1$, so we have a "canonical" representation of the inverse image functor $\textbf{Set} \to \textbf{Set} \times \textbf{Set}$ by a morphism of sites, namely the map $\{ \emptyset, \{ * \} \} \to \{ \emptyset, \{ a \}, \{ b \}, \{ a, b \} \}$ sending $\emptyset$ to $\emptyset$ and $\{ * \}$ to $\{ a, b \}$. It is not at all obvious to me how to extract anything resembling $\mathbb{Z}$ from this morphism of sites. In the first place, the "canonical" generator of $\mathbb{Z}\textbf{-Set}$ – $\mathbb{Z}$ with its self-action – is mapped to an object in $\textbf{Set}$ not even in the site we chose. In the comments I speculated that taking the limit of the diagram of sites would be a candidate site, but this example shows it does not work.