In ordinary representation theory over $\mathbb{C}$, all the irreducible modules of a finite group $G$ appear as composition factors of the tensor products $X \otimes \cdots \otimes X$ of a faithful $\mathbb{C}$-representation $X$. This is e.g. Thm. 10.8 in Ch.V of Huppert's Book "Finite Groups I".

This should also be true in positive characteristic, so let $X:G \to \mathrm{GL}(n,K)$ with $\mathrm{char}(K) = p$ be a modular faithful representation. Then all irreducible $KG$-modules appear as composition factors of the tensor products $X \otimes \cdots \otimes X$.

I'm not sure, how to prove this, but I think it shouldn't be difficult.