Ok I think I know how to do it. Its tricky because the isomorphism does not seem to be at all unique. that makes it not really stick out for you.
you have
an open cover by Ui, and on each overlap you have two clutching data, sij and tij, which are maps into the group of automorphisms of the fiber F.
then you need isomorphisms, i.e. homeomorphisms fi:Ui>Ui, such that
on overlaps, sij o fj = fi o tij. so you take f1 to be anything, then you need f2 = sij^(1) 0 f1 o tij on Ui meet Uj. so define it that way on Ui meet Uj and use an extension lemma like urysohn, to extend f2 to all of U2.
continue with f3,......
it seems compactness is needed. you could look it up in atiyah's book on k theory, or husemollers book on bundles, or steenrod, or any other book on bundles, like hatcher's algebraic topology, which is free online.
