A 1996 Jeep Grand Cherokee will not go into reverse if the transmission is defective or the shift linkage is not aligned properly. A local transmission shop or dealership can thoroughly diagnose and repair the issue.
On top of the transmission output. The best way to get to them is usually to remove the boot from the transmission shift tower, and go in through the floor.If you have an Allison automatic,, the reverse lamps are controlled through the transmission ECU.
Is it an automatic? Time for a new transmission... If a stick and 1st, 3rd, 5th are ok, then you have a possible shift cable problem. You should be able to see them on top of the trans. If everthing is moving properly the trans has an internal problem.
Diagnose first. If 1996 to present, an OBDII scan might help point you in right direction, if the ECU has detected a fault, for example a shift solenoid.. Could be anything as easy as adjusting linkage, to a solenoid, to as serious a needing a complete re-build
The Neutral safety switch is located on the back of the automatic transmission. it is quite simple to replace. It is probably easier to replace from under the vehicle, but it is accessable from the top if your vehicle is not equipped with ABS.