The simplest case of this question is if the bundle is trivial. So I think you have to accept as a building block the cohomology with trivial coefficients of your space and subspaces. But if you accept that, general cohomology can be computed in terms of these building blocks. For an open cover $X=U\cup V$, there is a Mayer-Vietoris sequence
$$H^i(X;L)\to H^i(U;L)\oplus H^i(V;L)\to H^i(U\cap V;L)\to H^{i+1}(X;L)$$
I let $L$ stand for "local system," but it works as well for a coherent sheaf. If it is trivial on $U$ and $V$, then those terms are your basic building blocks (more precisely, they are only isomorphic and you have to pay attention to the isomorphisms). The transition functions show up in the restriction maps from $U$ and $V$ to the intersection. You might think of the trivialization of $L$ on the intersection as coming from the trivialization on $U$, in which case the restriction map from $U$ is the usual map on cohomology with trivial coefficients. But the restriction map from $V$ is the composite of that map with the transition function.

If you have more than two open sets in your cover, then you can iterate this procedure, but the restriction maps are no longer between cohomology with trivial coefficients and cannot be identified with transition maps. But you can compute what they are at the previous stage.

This is probably the way to go for local systems. For vector bundles, it is not so nice, since you're probably trying to compute a finite dimensional vector space on a complete variety, but the open sets are not complete and thus have infinitely dimensional spaces of sections. But if the bundle is given by transition functions, this may be your only choice.