I think it would be enough to show that E*-B is locally contractible. But why is that?
I have problems to proof this one:
Let Y be a closed subset of a manifold Q, a vector bundle and E* the one-point compactification of E. Then which agree on Y are homotopic relative Y.
It is clear that E*-B is contractible. From the homotopy extension theorem we can extend the homotopy (f=g on Y) to all of Q, but why can we choose it in a way that it starts at f and ends at g?