I'm not quite sure what you mean by "with the exception of the wing tips". All finite wings generate vortices. So, yes, your purely rectangular wing will generate vortices.

If you're referring to the spanwise vortices that form along the leading edge of a highly swept wing at high angles of attack (like what's seen over the chines of an F-18 during a high-G maneuver), then no, a straight rectangular wing will not generate these vortices.