A discussion of the kind you want indeed seems to be difficult to locate. I am not an expert but I guess one could prove your claim (and more broadly, some version of the Stokes theorem) for non-orientable manifolds by passing to the two-sheeted covering oriented manifold, as suggested, in a slightly different setting, in the book Geometry VI: Riemannian geometry by Postnikov (here is the link to the relevant page on Google preview).
EDIT: As explained in the Bott--Tu book (see the link in Will's answer and also these two pages), rescuing the Stokes theorem in the non-orientable case requires passing from differential forms to densities.
