As pointed out, the answer is no – because we cannot hope to have fibrant replacements in general. On the other hand, by following the programme of van Osdol [1977, *Simplicial homotopy in an exact category*], it is possible to quickly establish a weaker result:

**Theorem.** For any regular category $\mathcal{S}$ (e.g. an elementary topos), the category of internal Kan complexes in $\mathcal{S}$ is a category of fibrant objects where the fibrations are the internal Kan fibrations and the trivial fibrations are the internal trivial Kan fibrations. (By Ken Brown's lemma, this suffices to determine the weak equivalences.)

Here, "internal" refers to the internal logic of regular categories: for example, an internal trivial Kan fibration in $\mathcal{S}$ is a morphism between simplicial objects in $\mathcal{S}$ such that the matching morphisms (à la Reedy) are regular epimorphisms. Note however that we are using the "external" notion of simplicial objects.

In the special case where $\mathcal{S}$ is a sheaf topos, the internal Kan fibrations and internal trivial Kan fibrations turn out to be the same thing as Jardine's local fibrations and local trivial fibrations. (See Theorem 1.12 in [1987, *Simplicial presheaves*].) It follows that the weak equivalences in the sense above are the same as Jardine's local weak equivalences.