Do simplicial objects in a Topos form a model category? Sometimes people say "If you don't like the word 'topos', just think the category of Sets", but I'm not sure to what extent this analogy holds. 
The real question here is, do simplicial object in a topos have the structure of a model category? I'm just not sure if you really need the geometric realization functor (and topological spaces) to define the usual model category structure on simplicial sets.
 A: As has been established in the comments the answer for a general topos is no, while for a Grothendieck topos it is yes, by the work of Joyal. 
The general question of when one can transfer a model structure on a category based on Sets to an arbitrary Grothendieck topos is beautifully adressed in Tibor Beke's articles on "Sheafifiable Homotopy Model Categories", available here. The examples include simplicial objects, cyclic objects and groupoid and category objects.
The proofs really use the assumption that you are in a Grothendieck topos (e.g. he chooses a site defining the given topos and uses the existence of morphisms to the topos of sets, plus the existence of the necessary colimits to interpret infinitary geometric logic, possibly even accessibility somewhere) and I don't think his arguments can be saved for more general toposes. Anyway the papers are worth a look and contain several interesting remarks about the role of accessibility in establishing model structures, giving a good perspective on your original question.
A: 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.
