The answer is yes. This follows from the fact that the group ${\mathbb Z}[S]$ should satisfy a universal property:
Let $A$ be any topological Abelian group and let $f: S \to A$ be a continuous function. Then $f$ can be extended uniquely to a continuous homomorphism ${\mathbb Z}[S] \to A.$
Suppose $G$ and $H$ are two copies of ${\mathbb Z}[S]$ with possibly different topologies. $G$ and $H$ both contain copies of $S.$ Applying the above unviersal property in two directions allows us to see that the identity from $G$ to $H$ is a homeomorphism.