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 might be: 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. You can find a proof for the $C^\infty$ setting in the book Nestruev, Smooth manifolds and observables.
Added in response to the comment:
You wrote:
Serre-Swan's theorem gives you that $\mathcal{S}^k$ can be identified with $\Gamma^\infty(M,V)$ for some vector bundle $V$ (provided we can show that $\mathcal{S}^k$ is finitely generated and projective).
On the other hand, here we would like to have a concrete description of $V$ as $U$ where the fiber over $x$ is $\mathcal{S}^k/I_x\mathcal{S}^k$.
That concrete description always follows from an identification $\Gamma^\infty(M,V)=\mathcal{S}^k$. To be precise: from a vector bundle $V$, a $C^\infty(M)$-module $\mathcal{P}$ and a $C^\infty(M)$-module isomorphism $\phi: \Gamma^\infty(M,V)\to \mathcal{P}$, we always obtain a natural identification $V_x= \mathcal{P}/I_x\mathcal{P}$. This identification is constructed as follows: given $x\in M$, let's consider $\mathbb{R}$ as a $C^\infty(M)$-module via the canonical identification $C^\infty(M)/I_x = \mathbb{R}$. Next we have a natural identification
$\mathcal{P}/I_x\mathcal{P}=\mathcal{P}\otimes_{C^\infty M} \mathbb{R}$, given in the direction
$
\mathcal{P}/I_x\mathcal{P}\to \mathcal{P}\otimes_{C^\infty M} \mathbb{R}
$
by
$$
p \text{ mod } I_x\mathcal{P}\mapsto p\otimes 1
$$
and in the other direction by
$$
p\otimes c \mapsto p\cdot c \text{ mod } I_x\mathcal{P}.
$$
(I skip the details of verifying that these maps are well defined and yield the identity when composed. Feel free to ask if something is not clear.)
Hence, tensoring the isomorphism $\phi$ with $\mathbb{R}$ we obtain an identification
$$
\alpha: \Gamma^\infty(M,V)/I_x\Gamma^\infty(M,V) \to \mathcal{P} /I_x\mathcal{P}.
$$
As last step, it's not hard to see that $\Gamma^\infty(M,V)/I_x\Gamma^\infty(M,V)$ is naturally identified with $V_x$
for any vector bundle $V$ (Corollary 11.9 in Nestruev).
To see that your concrete module $\mathcal{S}^k$ is finitely generated and projective we can construct an isomorphism $\mathcal{S}^k = \mathcal{Q}^*\otimes_{C^\infty(M)} \mathcal{Q}\otimes_{C^\infty(M)} S^k D$, where I'm using the following abbreviations
- $\mathcal{Q} =\Gamma^\infty(M,E)$
- $D$ denotes the module of vector fields on $M$, i.e. sections of the tangent bundle,
- $S^k$ denotes the $k$-th symmetric product of a $C^\infty(M)$-module.
- $\mathcal{Q}^*=\text{Hom}_{C^\infty M}(Q,C^\infty M)$,
In one direction this iso is defined by
\begin{align}
\mathcal{Q}^*\otimes_{C^\infty M} \mathcal{Q}\otimes_{C^\infty M} S^k D &\to \mathcal{P}^k/\mathcal{P}^{k-1}\\
r\otimes s \otimes (X_1\cdots X_k) &\mapsto s\circ X_1\circ\ldots\circ X_k \circ r \quad \text{mod} \quad \mathcal{P}^{k-1}
\end{align}
To make sense of the composition with $s$, use the canonical identification of a module $\mathcal{Q}$ with $\text{Hom}_{C^\infty M}(C^\infty M, \mathcal{Q})$.
The other direction is also not to hard to define, but slightly more involved: start with the observation that given $(\nabla \text{ mod } \mathcal{P}^{k-1}) \in \mathcal{P}^k/\mathcal{P}^{k-1}$ and $a_1,\ldots,a_k \in C^\infty M$, then $[\cdots [\nabla,a_1],a_2]\cdots ,a_k]\in \mathcal{Q}^*\otimes \mathcal{Q}$ is well-defined (independent of the representative $\nabla\in \mathcal{P}^k$) and multilinear symmetric and a derivation in the $a_i$'s. From there you can use the universal derivation $d: A \to \Lambda^1$ (differential one forms) to show that you get a well defined map from $ \mathcal{P}^k/\mathcal{P}^{k-1}$ to $\mathcal{Q}^*\otimes_{C^\infty(M)} \mathcal{Q}\otimes_{C^\infty(M)} S^k D$.
This identification not only shows that $\mathcal{S}^k$ is projective and finitely generated, it gives a more concrete description of fibers $\mathcal{S}^k/I_x\mathcal{S}^k$: they are symbols $S^k T_xM\otimes_\mathbb{R} \text{Hom}_\mathbb{R}(E_x,E_x)$, as mentioned in Peter Michor's answer.
