$\mathbf{Log}$, the category of elementary topoi and logical morphisms *which preserve everything on the nose* is cocomplete; in

Eduardo J Dubuc, GM Kelly - *A presentation of topoi as algebraic relative to categories or graphs* - Journal of Algebra (website)

this category $\mathbf{Log}$ is shown to be the category of algebras of a finitary monad on $\mathbf{Cat}$; it looks like you can even get this over $\mathbf{Grph}$, like the presentation of cartesian closed categories monadic over graphs in Lambek-Scott intro to higher-order categorical logic. This implies then that it is cocomplete, see the nLab page on colimits in categories of algebras, for example.

For the sake of completeness, a nice account (including explicit constructions) of 2-limits in $\mathbf{Log}$ as a locally groupoidal 2-category (1-cells the standard notion of logical morphism, 2-cells between them restricted to be isomorphisms) is given in chapter III of Steve Awodey's PhD thesis,

S Awodey - *Logic in topoi: functorial semantics for higher-order logic* - available from his website