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In the paper Affine varieties dominated by $\mathbf{C}^2$ Gurjar considers a slightly more general situation, namely an affine normal variety $\mathrm{X}$ with a proper surjective morphism $\mathbf{A}^2\rightarrow\mathrm{X}$. He shows that every line bundle on $\mathrm{X}$ is trivial; together with a result of Anderson (every vector bundle on $\mathrm{X}$ is the direct sum of a trivial bundle and a line bundle) this shows in particular that every vector bundle on $\mathbf{A}^2/\mathrm{G}$ is trivial.