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This is an attempt to complete Tyler's argument. We first note that $K^0(S^5)=\mathbb Z$ (note there is a parity difference, $K^*(S^n)$ is always $\mathbb Z^2$ but for $n$ odd we have $K^1(S^n)=\mathbb Z$ and hence $K^0(S^n)=\mathbb Z$). This means that every topological vector bundle on $S^n$ is topologically stably trivial. Let now $E$ be an algebraic vector bundle on $S^n$, i.e., an f.g. projective module over $\mathbb R[x_0,\dots,x_n]/(x_0^2+\cdots+x_n^2-1)$, of rank $k$. As it is topologically stably trivial that means that there are continuous sections $f_1,\dots,f_k$ of some $E\bigoplus R^m$ trivialising it, i.e., form a basis at each fibre. Now, being a trivialisation is an open condition under the sup norm (with respect to some metric on the vector bundle to be precise) so if we can show that the algebraic sections of any vector bundle $F$ are dense in the space of continuous sections we get that it is also algebraically trivial. However, picking a $G$ such that $F\bigoplus G$ is trivial reduces this to showing that $\mathbb R[x_0,\dots,x_n]/(x_0^2+\cdots+x_n^2-1)$ is dense in the ring of continuous (real-valued) functions on $S^n$ but this follows from the Stone-Weierstrass theorem.