Solving modular matrix equations via Gaussian elimination or System of linear equations (SIS assumption?)
Let $A' := [A_1 ~ | ~ \dots ~ | ~ A_k ] \in \mathbb{Z}_q^{m \times n k}$
and
$C' := [C_1 ~ | ~ \dots ~ | ~ C_k ] \in \mathbb{Z}_q^{m \times n k}$.
Then define $A \in Z_q^{m\times m}$ as the matrix whose columns are the first $m$ columns of $A'$. Likewise, define $C$ using the first $m$ columns of $C'$.
Then, $S$ satisfies $S \cdot A = C \pmod q$, thus, you can find it by computing $A^{-1}\cdot C \pmod q$.

Side note: why do you think that this problem is similar to the SIS problem? Notice that the SIS problem has a parameter $\beta$ that is an upper-bound to the norm of the accepted solutions and Gaussian elimination is likely to output solutions whose norm is above $\beta$ (thus, not valid solutions).