reference request - Is every real vector bundle over the circle necessarily trivial?

In the spirit of Theo's comment, I'll say something about sufficient conditions.

A real vector bundle over the circle is trivial if and only if it is orientable. I discussed this a bit here.

The main point is that up to isomorphism, every real vector bundle over the circle is either trivial, or the Whitney sum of a trivial bundle with the Mobius bundle. The latter is not orientable.

The other answers at the question I linked to above may also be helpful to you.