Let $R$ be a ring object in a elementary topos $X$. Does the category of $R$-modules in $X$ possess enough injectives? If $R$ is a Grothendieck topos, this is a well-known fact, and it is evident in any other examples I can think of, e.g. the topos of sheaves of finite sets on a finite site. The category of $R$-modules does not form a Grothendieck category in this case, lacking arbitrarily large direct sums.
Edit: as an important caveat to point out, the category of Abelian groups in $X$ does not have enough injectives. For example in the case $X$ is the topos of finite sets, the category of finite Abelian groups doesn't have enough injectives. If the original claim is true, it is critical that we structure over a ring internal to $X$.
Best Answer
No. For instance, Andreas Blass proved in
that there are models of ZF in which no nontrivial abelian groups are injective, and such a model is then an elementary topos with a ring $\mathbb{Z}$ whose modules do not have enough injectives.
(This is assuming ZF is consistent, of course, but you can eliminate that assumption if all you're looking for is an elementary topos, since for that you only need a model of a sufficiently large fragment of ZF, and that can be constructed in just ZF using the reflection principle.)