Multimeters frequently come in handy for me, and they cost very little, probably less than $15 at Radio Shack. I would also strongly urge you to purchase the appropriate shop manuals for your model year, and an AIM (Assembly Instruction Manual) can be very handy. You'll find them at NCRS.org, and many of the Corvette parts vendors carry them.
On looking at a parts manual, it appears there are two studs on the backside of each of the two backup lamp fixtures, almost directly behind the lense screws, that are used to hold the fixtures to the fiberglass with nuts. One stud of each fixture should probably have a grounding wire attached under the nut.
Since both fixtures are grounded, the multimeter test I mentioned previously will tell you if the bad side is grounded... try the test between the socket inner walls first, then betwwen the fixture castings themselves, then between each socket wall and it's fixture casting; if the castings are both grounded but the "bad" socket is not, you may just be able to wedge a couple of pins between the socket and the casting from the outside to restore connectivity to ground.
If you can't reach up behind the rear exhaust valence panel to feel around and/or remove the stud nuts with an ignition box wrench, then there are 2 other possibilities:
1. Try removing the license light fixture (held in by screws) to see if you can then see or reach the backside of the backup light fixtures from there.
2. If not, you may be forced to remove the lower rear exhaust valence panel to gain access room... this is the panel the exhaust pipes pass through, and it is held in place at the top by screws or bolts into the panel that holds the tail lights (the fastener heads are hidden from view by the bumpers) and at the sides there are bolts holding this panel to the rear quarter (fender) panel on the backside. I can't recall, but I think you may first need to remove the tailpipe trim bezels, the chrome extension pipes, and maybe even the bumpers, though, in order to loosen the exhaust panel upper screws and wiggle the exhaust panel down and out.
