This concept appears in Bott&Tu's GTM82. A flat vector bundle is one who has a particular trivialization with locally constant transition functions. Then my question is whether every vector bundle over a manifold admits such a trivialization.
btw: Are the tags correct?
No. Using the Chern-Weil perspective on characteristic classes (see here), you can prove that all the rational Pontryagin classes of a flat vector bundle have to vanish. Thus all you need are vector bundles with non-vanishing rational Pontryagin classes, of which there are many.
A very nice source for this perspective on characteristic classes and flat bundles is Morita's book "Geometry of characteristic classes".