Is there anything that can be said in general about the Cech cohomology of the sheaf $\mathcal{F}$ of sections of a holomorphic bundle over a contractible space $X$?

I know that such a bundle must be trivial, but I don't know much about Kunneth formulae, so haven't made much progress on that front. I've also thought about embedding it in an exact sequence involving the sheaf $\mathbb{Z}$ over $X$, but this doesn't seem to lead to any general results.

Could someone give me a hint as to whether there's anything obvious I'm missing? Otherwise just let me know that I'm looking for something non-existent! This is motivated by a desire to use the Mayer-Vietoris sequence to show that some cohomology groups are trivial by splitting the base space into contractible pieces - perhaps this isn't a method which works in general?

Many thanks in advance!