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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 wouldn't be too optimistic that don't think his arguments can be saved for much more general toposes(you would probably need a geometric morphism to the topos of sets plus the necessary colimits to interpret geometric logic. If it then turns out that accessibility is needed somewhere then you are almost back to Grothendieck 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.

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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) and I wouldn't be too optimistic that his arguments can be saved for much more general toposes (you would probably need a geometric morphism to the topos of sets plus the necessary colimits to interpret geometric logic. If it then turns out that accessibility is needed somewhere then you are almost back to Grothendieck 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.