Find collision in Ajtai's hash function using short vector
It depends on the exact domain of the hash function and the quality of the SIS solution, i.e., its norm (and the choice of norm itself).

Suppose the hash domain is $\{-d, \ldots, d\}^m$, i.e., vectors of $\ell_\infty$ norm at most $d$. Then any nonzero vector in Ajtai's lattice having Euclidean norm at most $d$ collides with the all-zeros input. (Actually, $\ell_\infty$ norm at most $d$ suffices.) Thus, finding short enough lattice vectors yields collisions.

But now suppose $d=1$, say. (This is a typical choice.) For standard parameters, the shortest (in Euclidean norm) nonzero vectors in Ajtai's lattice have norm about $\sqrt{n \log q} \gg d = 1$.  Thus, short vectors in Euclidean norm (which is what typical lattice-basis reduction algorithms deliver) may not be sufficient to yield a collision.  Instead, finding a collision requires finding a shortest nonzero vector in the $\ell_\infty$ norm. But little is known about lattice-basis reduction algorithms for this setting.

More details on these issues can be found, e.g., in the paper "SWIFFT: A Modest Proposal for FFT Hashing" (I am a coauthor).