I'm interested in examples of sphere bundles which do not arise from vector bundles.
I'm not quite clear about the following. So please let me know if anything is false.
I believe that a $(k-1)$-sphere bundle arises from a vector bundle iff its structure group can be reduced to $O(k)$. I think that for $k\geq 4$ it is not known if $O(k)$ is homotopy equivalent to $\operatorname{Diff}(S^{k-1})$, and in general it is false. So there are sphere bundles that do not arise from vector bundles.
I'm also interested in more details/clarification of this argument.