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The vector space $U_x$ will be infinite dimensional, so it's not immediately clear what $\Gamma^\infty(M,U)$ denotes. I assume you mean $\Gamma^\infty(M,U):=\bigoplus_k \Gamma^\infty(M,U^k)$ where $U^k_x:=\mathcal{S}^k/I_x\mathcal{S}^k$. Then the question reduces to: Why is $\mathcal{S}^k$ isomorphic to $\Gamma^\infty(M,U^k)$? When $\mathcal{S}^k$ is projective an finitely generated over $C^\infty(M)$ (which I think it is in your setting) the answer is the Serre-Swan theorem. I think you can find a proof for the $C^\infty$ setting in the book Nestruev, Smooth manifolds and observables.