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 using bundles of the form $\oplus_{g \in G} g^* \mathcal O(n)$. Other than the "averaging trick," the construction of this resolution is identical to Serre's non-equivariant resolution, i.e. twist high enough that your sheaf is globally generated, use a surjection from a trivial bundle, twist back down, and then repeat for the kernel.
I've used the finiteness of $G$ and the projectivity of $X$ heavily, and I'm not sure what you can say without them.