This is worked out in part 2 of
Adams, J. F., Stable homotopy and generalised homology, Chicago Lectures in Mathematics. Chicago - London: The University of Chicago Press. X, 373 p. £ 3.00 (1974). ZBL0309.55016.
(note that to understand part 2 you need to have read part 3 first. Yeah, I know)
In particular the result you want is true only for complex orientable cohomology theories (those theories for which we can define $c_1$). This is to be expected, since what you want is essentially a version of the Thom isomorphism, that should hold only if the vector bundle is orientable for your cohomology theory.