The hardness of SIS (or more precisely, the hardness of the module version of SIS: MSIS) is the assumption used to demonstrated the Strong Unforgeability of Crystal Dilithium under Chosen Message Attacks (SUF-CMA). In other words if this problem is hard then even producing a new signature for a previously signed message is hard even if the adversary is allowed to request multiple signatures of other messages. This is seen in section 6.2.2 of the paper.
If one can solve MSIS, powerful attacks are possible: consider the key generation algorithm on page 4 of the specification:
Gen
01 $A←R^{k×l} q$
02 $(\mathbf s_1,\mathbf s_2)\leftarrow S_\eta^l \times S_\eta^k$
03 $\mathbf t := A\mathbf s_1 +\mathbf s_2$
04 return(pk=($A,\mathbf t$),sk=($A,\mathbf t,\mathbf s_1,\mathbf s_2$))
We are given the values $A$ and $\mathbf t$ and wish to recover $\mathbf s_1$ and $\mathbf s_2$ for a key recovery attack (top of the attack outcome hierarchy).
Note that the matrix
$$\begin{matrix}(A|I|-T)\end{matrix}$$
where $I$ is the $k\times k$ identity matrix and $T$ is the $k\times k$ diagonal matrix with entries taken from $\mathbf t$ has the short solution $(\mathbf s_1^T,\mathbf s_2^T,1,\cdots 1)^T$. A solution to this problem therefore allows recovery of the private key and complete impersonation of the owner.
