As is stated for example in this post (Structures on Vector Bundles with Reduced Structure Group), we can impose interesting restrictions on real vector bundles by demanding their transition functions to lie in certain subgroups of $\operatorname{GL}(n)$; for example if they are contained in $\operatorname{GL}^+(n)$, the respective bundles must be orientable; if they are in $\operatorname{U}(n)\subset \operatorname{GL}(2n)$, they are complex vector bundles with a Hermitian scalar product, and for $\operatorname{SU}(n)\subset \operatorname{GL}(2n)$, the determinant bundle must even be trivial.
However, we can also restrict our attention to vector bundles with transition functions in one of the more "exceptional" subgroups $G_2 \subset \operatorname{GL}(7)$ and $\operatorname{Spin}(7) \subset \operatorname{GL}(8)$. Are there also such nice descriptions of these bundles? According to Wikipedia, a manifold $M$ has a tangent bundle with transition functions choosable in $G_2$ iff it is a $7$-dimensional Spin-Manifold, but I couldn't find any literature on the general case. Has anyone else encountered something that could help?