If I am correct, the 12 power is sent to the backup lights from the neutral safety switch, which is found on the driver side of the trans. I believe the NSS gets its 12V feed from a key-on ignition source - there should be 3 wires at the NSS - one to the backup lights, one a 12V feed, and the third being the grounding wire from the the starter relay, on the ANGLED connector tab found on the bottom corner of the relay (found on the firewall or inner fender). The 12V feed CANNOT flow to the backup lamps if the comb inside the transmission is not pushing the ball into the switch and completing the circuit. There are 3 steps to the ball's positioning based on detents/bumps on the comb - see below.
1 will not engage the switch, so no current flow anywhere, and the car won't start. This will be in every position BUT reverse.
2 is going to give you current flow to the backup lights, but NOT ground out the start relay to permit the starter to engage.
3 is going to engage the relay only to permit the starter to engage, but not send 12V to the backup lamps.
Unless you're in Park or Neutral, the comb will not engage the NSS, and no start and no 12V to the backup lights.
There will be 2 identical steps on the comb - Park and Neutral.
There will be 1 unique detent on the comb - reverse lights. This is the ONLY position that 12V will be found on 2 wires (if the switch is working properly).
I think I have that right, but others will please chime in to correct me. Regardless, look for power coming to the switch from a key on source, a wire running to the start relay, and a wire running to the backup lamps.
Putting 12V to the wire running from the Neutral Safety Switch to the backup lamps should make them light up, provided the lamps have a good ground at the housing.
Connecting the key on hot 12V feed to the wire running to the backup lamps (bypassing the NSS) should make them light up, provided the lamps have a good ground at the housing. If this happens, then you likely need a new NSS.