I previously indicated how to construct the tensor product for $\textbf{Ab}(\mathcal{V})$ when $\mathcal{V}$ is a sufficiently nice cartesian closed category. (The comments are probably more helpful than the post itself.) A Grothendieck topos is certainly nice enough, but so is an elementary topos with a natural numbers object. (See Theorem D5.3.5 in *Sketches of an elephant*.)

Basically, we mimic the standard construction of tensor products by generators and relations; the only stumbling block is the construction of free internal abelian groups. Once this is done, $\textbf{Ab}(\mathcal{V}) \to \mathcal{V}$ will be monadic, say with left adjoint $F : \mathcal{V} \to \textbf{Ab}(\mathcal{V})$, and $\textbf{Ab}(\mathcal{V})$ will have coequalisers for reflexive pairs. (See Lemma D5.3.2 in *Sketches of an elephant*.) One can then define a certain object $I$ in $\mathcal{V}$ for each pair of internal abelian groups $A$ and $B$ and a pair of morphisms $I \rightrightarrows F (A \times B)$ in $\mathcal{V}$ such that
$$F I \rightrightarrows F (A \times B) \to A \otimes B$$
presents $A \otimes B$ as a coequaliser of a reflexive pair in $\textbf{Ab}(V)$. (We could take
$$I = (A \times A \times B) \amalg (A \times B \times B) \amalg (F1 \times A \times B) \amalg (A \times F1 \times B)$$
so that we can code the four equations
\begin{align}
(a + a') \otimes b & = a \otimes b + a' \otimes b \newline
a \otimes (b + b') & = a \otimes b + a \otimes b' \newline
(r a) \otimes b & = r (a \otimes b) \newline
a \otimes (r b) & = r (a \otimes b)
\end{align}
but one can probably get away with less.)

It turns out to be much easier to construct internal homs for $\textbf{Ab}(\mathcal{V})$: you just need $\mathcal{V}$ to be cartesian closed with equalisers. Of course, once you have both the tensor product and internal homs, then they make $\textbf{Ab}(\mathcal{V})$ into a symmetric monoidal closed category.

One can also carry out this construction in much greater generality for any commutative monad on any sufficiently nice symmetric monoidal closed category $\mathcal{V}$. This, I think, is an old result of Kock [1971, 1972] generalising Linton [1966].