**No — in particular, not if $R$ has any non-integer rationals.**

Briefly: From $\newcommand{\Z}{\mathbb{Z}}\newcommand{\Ab}{\mathrm{Ab}}\newcommand{\RMod}{{R\text{-}\mathrm{Mod}}}\RMod$, we can recover $R$ as the ring of endomorphisms of the identity functor, $\newcommand{\End}{\mathrm{End}}\newcommand{\id}{\mathrm{id}}R \cong \End(\id_\RMod)$. But for a topos $\newcommand{\E}{\mathcal{E}}\E$, $\End(\id_{\Ab(\E)})$ retracts onto $\Gamma(\Z_\E)$ (global sections of the integers of $\E$); and this will never have any non-trivial rationals.

Unwinding this more elementarily: Suppose $n \in \Z$ is invertible in $R$, with $n \neq \pm 1$. Then multiplication by $\frac{1}{n}$ gives a natural endomorphism $\mu_{\frac{1}{n}}$ of the identity functor on $\RMod$, satisfying $n \cdot \mu_{\frac{1}{n}} = 1$ (where $1$ is the identity endomorphism of the identity functor).

But the $\mathbf{Ab}$-enrichment of any Abelian category is determined by the category structure (using the biproducts). So an equivalence $\RMod \cong \Ab(\E)$ would transfer $\mu_{\frac{1}{n}}$ to an endomorphism $\mu'$ of $\id_{\Ab(\E)}$ with $n \cdot \mu' = 1$; then applying this at $\Z_\E$, and taking global sections, would give an inverse for $n$ in $\Gamma(\Z_\E)$. But this can never exist unless $\E$ is the trivial topos, since as $n$ is non-unital you can prove constructively “$n$ is not invertible in $\Z$”, so this holds in the internal language of $\E$. So then $\Ab(\E)$ is also trivial, and the equivalence $\RMod \cong \Ab(\E)$ implies $R \cong 1$.

So we’ve shown: **if an integer $n \neq \pm 1$ is invertible in $R$, and $\RMod \cong \Ab(\E)$, then $R$ is the zero ring.**

noin general. $\endgroup$