Suppose we are given a vector bundle $p:E \to B$, we consider an open cover $\{U_\alpha\}_{\alpha \in A}$ of $B$ with local trivializations $h_\alpha:p^{-1}(U_\alpha) \to U_\alpha \times \mathbb R^n$.
Then, we can recover the total space by taking $\coprod_\alpha U_\alpha \times \mathbb R^n/\sim$, where $(x,v) \sim h_\beta h_{\alpha}^{-1}(x,v) \in U_{\beta} \times \mathbb R^n$, whenever $x \in U_\alpha \cap U_{\beta}$.
I want to find the appropriate diagram so that I can understand this construction as the pushout of a diagram along each of these maps $g_{\beta\alpha}:=h_\beta h_{\alpha}^{-1}$.
Does anyone know of a reference for this kind of treatment, or if it is even fruitful?
The quotient of an object $X$ by an equivalence relation $R\subset X\times X$, in any category with enough structure, is defined as the coequalizer of the two projections $R\stackrel{\to}\to X$. Any coequalizer can be written as the pushout of the corresponding span $X\leftarrow R\sqcup X\to X$, so in that sense your vector bundle is a pushout, but that's not a very natural viewpoint.
The most natural diagram to use here has, I think, a more complicated shape. Consider the poset structure on the set $S'=A\sqcup A\times A$ generated by $(\alpha,\beta)\leq \alpha$ and $(\alpha,\beta)\leq \beta$, for each $(\alpha,\beta)\in A\times A$. It is most natural, but immaterial, to also remove the diagonal of $A\times A$ from $S'$, yielding finally a poset $S$. When $A$ contains two points, this construction yields the poset $\alpha\leftarrow (\alpha,\beta)\rightarrow \beta$, so that colimits along $S$ for larger index sets $A$ are generalized pushouts.
Now your vector bundle is the colimit of the $S$-indexed diagram of spaces sending $\alpha\mapsto U_\alpha\times \mathbb{R}^n$, $(\alpha,\beta)\mapsto p^{-1}(U_\alpha\cap U_\beta)$, and the comparison maps $(\alpha,\beta)\leq \alpha,(\alpha,\beta)\leq \beta$ to the restrictions of $h_\alpha$ and $h_\beta$, respectively. This colimit has exactly the desired effect of identifying $(x,v)$ with $h_\beta h_\alpha^{-1}(x,v)$ when sensible.
It might be worth remarking that the same story describes how to describe a manifold as a colimit of its coordinate patches, and for many other situations in which objects are built by gluing together local data.