The following fact is mentioned in the book of Huppert-Blackburn vol. 2.
Let $G$ be a finite group and $K$ a field of characteristic $p$. Then there are only finitely many indecomposable $K[G]$-module if and only if Sylow-$p$- subgroups of $G$ are cyclic.
So considering this fact, I was in search of examples.
For the group $C_3\times C_3$, what is (natural) infinite family of indecomposible modules with field $\mathbb{F}_3$?
Let $g$ and $h$ be generators of $C_3\times C_3$.
Then, for example, for each $n>0$ there is a $2n$-dimensional indecomposable module on which $g$ acts by $\pmatrix{I_n&I_n\\0&I_n}$, where $I_n$ is the $n\times n$ identity matrix, and $h$ acts by $\pmatrix{I_n&J_n\\0&I_n}$, where $J_n$ is the $n\times n$ Jordan block with eigenvalue $1$.