I have a 2006 GX470. On Sept 1 and Oct 1 it would start and run, but the shifter would not move to reverse. I checked the OBD and there was no power. The following lights were on: Brake/ABS/airbag/differential lock/check engine/tire pressure.
The first time it happened I had it towed to lexus dealer. When it arrived there were no issues they could find.

Today it happened again-I have driven this vehicle everyday hoping it would happen so I could diagnosis it. I went to get a needle nose plier to pull a fuse and when I got back to the car it was fine-no issues.

I noticed that the AC INV relay housing is cracked. I will replace it. Any ideas??? It was cool, but not damp both times this has happened.

 

Check your brake switch. I had a similar issue, and it was due to a faulty brake switch.