Write \(X=\operatorname{Spec} A\). The action of \(G\) on \(X\) is equivalent to a coaction \(A\to A\otimes k[G]\), so \(A\) is a rational \(G\)-module. Choose finitely many algebra generators of \(A\). The finite-subcomodule argument used in the embedding theorem puts these generators inside a finite-dimensional \(G\)-stable subspace \(W\subseteq A\).

The inclusion \(W\subseteq A\) gives a surjective algebra map \(\operatorname{Sym}(W)\to A\), hence a closed immersion \(X\hookrightarrow W^\vee\). Since \(W\) is \(G\)-stable, the dual space \(W^\vee\) is a finite-dimensional representation and the immersion is \(G\)-equivariant.