There is a very useful slogan that those constructions involving vector spaces which are canonical (does not require any choices) does carry on to vector bundles as well. I would like to confront this with one particular example: let $E$ be a vector bundle over $X$ and $V$ be a subbundle of $E$. We can form the quotient bundle $E/V$. My question is:
Do we have an isomorphism $E \cong V \oplus E/V$? If so, does this isomorphism is canonical?
I assume you are in the differentiable category. Let $g$ be a differentiable metric on $E$, you can define $V'$ the orthogonal bundle of $V$. The restriction of the canonical projection $p:E\rightarrow E/V$ to $V'$ is an isomorphism.
In the complex category, this is not necessarily true, there exists an obstruction in $H^1(X,)$ which is the obstruction of such a splitting.