Suppose $L$ on $X$ is a (very) ample line bundle. Then $\oplus_{g \in G} g^* L$ is a $G$ equivariant vector bundle. Its determinant is also $G$ equivariant and isomorphic to $L^{\otimes \# G}$, thus (very) ample.
Now this line bundle gives us a $G$-equivariant embedding into projective space, and to construct the resolution by $G$-equivariant vector bundles we can construct a resolution on projective space using bundles of the form $\oplus_{g \in G} g^* \mathcal O(n)$ and then pull back to $X$. Other than the "averaging trick," the construction of this resolution is identical to Serre's non-equivariant resolution.
I've used the finiteness of $G$ and the projectivity of $X$ heavily, and I'm not sure what you can say without them.