They're illegal to drive with (ie won't pass inspection), not illegal to have (show car is ok).
19VAC30-70-250. Doors.
Inspect for and reject if:
1. If each door located at the left and right side of the driver's seat is not equipped with an opening device similar to that installed by vehicle manufacturers which will permit the opening of the door from the outside and inside of the vehicle.
2. If each door located to the left and right side of the driver's seat is not equipped with a latching system similar to that installed by vehicle manufacturers which will hold the door in its proper closed position.
http://leg1.state.va.us/cgi-bin/legp...19VAC30-70-250