In Milnor's book, Topology from the Differentiable Viewpoint, the definition of the index of a smooth vector field with isolated zeros on a manifold does not seem to require that the manifold is orientable. The definition is local i.e. defines the index around each zero in a small open ball.
So it seems then that the Poincare-Hopf theorem can be proved for an unorientable manifold even though the references I've found through Google state that the manifold is orientable.
The proof extends the vector field to a tube around the manifold so that the extended vector field points outward along the boundary of the tube and is homotopic to the Gauss mapping of the boundary into the unit sphere. I think that the sum of the indices of the vector field on the manifold ,even if it is not orientable ,therefore equals the degree of the Gauss mapping because the tube is orientable.
Is there a mistake in this reasoning?
