Unfortunately, the problem as stated is not quite restrictive enough to meaningfully exploit. For any "interesting" sparse matrix $\mathbf A$, even if the forcing data $\mathbf b$ is sparse, the solution data $\mathbf x$ will still be fully populated/dense.
However, there is a nearby problem that does have exploitable structure: if $\mathbf b$ is sparse, and you only care about a subset of the entries of $\mathbf x$, then it is possible to find them using less time/space than the naive approach (in which you compute all of $\mathbf x$ and then sift out the ones you want).
Unfortunately this algorithm is not commonplace, but I can at least point you to my own implementation. I think MUMPS might have a similar capability, their names pop up often when you look for these sorts of tricks (sparsity-exploiting backsolves). The earliest mention of this idea that I have found is in a thesis by T. Slavova (who was advised by the MUMPS team).