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 $\otimes_{g \in G} {g^*L}$, with each factor  (very) ample.  Thus it is (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.