Vector-Valued Stone-Weierstrass Theorem?

This is a comment, not an answer but I am, alas, not entitled. Vector valued Stone-Weierstraß theorems were studied in great detail in the second half of the last century and there is a comprehensive monograph on the subject by João Prolla ("Weierstraß-Stone, the theorem", 1993). Not on topic, but he also considered the case of bounded, continuous vector-valued functions on non-compact spaces, using the strict topology of R.C. Buck.

I think that you want something like this:

Let $Eto X$ be a (finite rank) vector bundle over a compact, Hausdorff topological space $X$, let $mathcal{A}subset C(X,mathbb{R})$ be a subalgebra that separates points, and let $mathcal{E}subset C(X,E)$ be an $mathcal{A}$-submodule of the $C(X,mathbb{R})$-module of continuous section of $Eto X$. Suppose that, at every point $xin X$, the set ${,e(x) | einmathcal{E} }$ spans $E_x$. Then $mathcal{E}$ is dense in $C(X,E)$ with respect to the sup-norm defined by any norm on $E$.

Addendum: Here is a sketch of the argument: First, by an easy compactness argument, one can show that $mathcal{E}$ contains a finite set $e_1,ldots e_m$ such that $e_1(x),e_2(x),ldots,e_m(x)$ spans $E_x$ for all $xin X$. Then $mathcal{E}$ contains all the sections of the form $$a_1, e_1 + cdots + a_m,e_m$$ where $a_iinmathcal{A}$, and every section $ein C(X,E)$ can be written in the form $$e = f_1, e_1 + cdots + f_m,e_m$$ for some functions $f_iin C(X,mathbb{R})$. By the Stone-Weierstrass Theorem, for any given $delta>0$, we can choose $a_iin mathcal{A}$ so that $|f_i-a_i|<delta$ for all $1le ile m$. Now the equivalence of all norms in finite dimensional vector spaces can be applied (together with the compactness of $X$) to conclude that $mathcal{E}$ is dense in $C(X,E)$ in any sup-norm derived from a norm on the (finite rank) vector bundle $E$.