Find combination of vectors from array that sum up to s
If $k$ is part of the input, the problem is NP-complete even when $m=1$ (and when we want to know if there's even a single solution). That is, when $m=1$, we essentially have a collection of $n$ values and our task is to decide if there is a subset of those that sum up to a target $s$. This is known as subset sum.

The Wikipedia page lists algorithms that you might be able to adapt to your more general case as well. Also, if $k$ is small enough, you might be able to do something better.