well i looked in bott tu p. 58 for the vector bundle case, and it is knid of like what i said.
one shows the isomorphism class of a bundle on YxI, restricted to Yx{0} cannot change locally near 0.
i.e. the homotopy gives a pull back bundle E on YxI, and we look at its restriction to Yx{0} and we just cross that restriction with I to get a family F of bundles on YxI, which have isomorphic restrictions to every Yxt.
So we have an isomorphism of these bundles E,F over 0 and want to extend the isomorphism say to Yxe where e is a small interval containing 0.
To do this view the isomorphism as a section defined over Yx{0}, of the subbundle Iso(E,F) of Hom(E,F).
Then since this last bundle has Eucklidean space fibers, we can extend, again by a Urysohn type lemma.
then near t=0 the section lies in Iso(E,F), and using compactness of Y and I we get our theorem over YxI.
It can be extended to paracompact Y. This argumenT uses that the bundle of isomorphisms of two vector bundies is itself a fiber bundle contained in a euclidean space bundle, which seems a little special for all fiber bundles..
to go further i guess we need to know your definition of a fiber bundle. but i will probably leave it here.
