As the title says, are there any relevant theorems, which determines when, given an abelian additive group $(M,+)$, we can determine whether $(M,+)$ admits a $R[G]$-module structure, for a unital ring $R$ and group $G$.
Are there any conditions on $R$ or $G$ that makes it easier to determine (for example $R$ commutative or $G$ finite)?
Let $(M,+)$ be an Abelian group. Recall that the unital left $S$-module structures on $M$ are in bijective correspondence with unital ring homomorphisms $\varphi:S\to\mathrm{End}_{\mathbb{Z}}(M)$.
Hence, if $(M,+)$ has a left $R[G]$-module structure then there exists a unital ring homomorphism $\psi: R[G]\to \mathrm{End}_{\mathbb{Z}}(M)$. In particular, the restriction $\varphi = \psi_{\mid_R}: R \to \mathrm{End}_{\mathbb{Z}}(M)$ is a ring homomorphism and hence yields a left $R$-module structure on $M$. Moreover, the restriction $\sigma = \psi_{\mid_G}: G \to \mathrm{Aut}_{\mathbb{Z}}(M)$ is a group homomorphism from $G$ to the group of $\mathbb{Z}$-automorphisms $\mathrm{Aut}_{\mathbb{Z}}(M)$ of $M$. And thirdly, since elements $a\in R$ and $g\in G$ commute in $R[G]$, we must have that the images $\varphi(a)$ and $\sigma(g)$ commute in $\mathrm{End}_{\mathbb{Z}}(M)$.
Conversely, if $M$ is a left $R$-module, say via $\varphi: R\to \mathrm{End}_{\mathbb{Z}}(M)$, and there exists a group homomorphism $\sigma: G\to \mathrm{Aut}_{\mathbb{Z}}(M)$ such that $$ \varphi(a)\circ \sigma(g) = \sigma(g)\circ \varphi(a), \qquad \forall a\in R, g\in G$$ then $$\psi: R[G]\to \mathrm{End}_{\mathbb{Z}}(M), \qquad \sum_{g_\in G} a_g g \mapsto \sum_{g_\in G} \varphi(a_g)\circ \sigma(g)$$ is a ring homomorphism, which defines a left $R[G]$-module structure on $M$.
To summarize: you need an $R$-module structure on $M$ and a group homomorphism $G\to \mathrm{Aut}_{\mathbb{Z}}(M)$ that commutes with the $R$-module structure in order to define an $R[G]$-module structure on $M$.