Jeep grand cherakee 1998 License plate lights not working.

You need to get a 12v test light and probe the harness to find out where the issue is.

The lights are on the same circuit with the side marker lights. So if those work, and you don't have power at the lic plate sockets, somethings going on with the wiring harness in the back of the truck.
If you have power the ground circuit may be damaged.

They are tied to the entire running/parking lamp circuit. I suspect a broken wire or a bad ground. Inspect the harness up where the liftgate hinges are for damage, Peel the wrap off and wiggle the wires and see if plate lamps flicker.
Or, pull the inside liftgate panel off as needed to see the wire colors. Using a test light, work your way up the harness on those color wires until you find power. You may find the lamp sockets are burnt and crispy.
Or, go get an aftermarket LED light(white light) and wire it into the running lamp circuit.
Hope this helps.

Check your gas cap for tighness / damage

LOOK LIKE ALL JEEP GRAND CHEROKEE FROM 1995 TO 2001 HAS THE SAME TRANSMISSION.BUT HAS DIFFERENT TRANSMISSION CONTROL MODULES.SO I WOULD BUY EXACT YEAR MODEL VECHICLE TRANSMISSION.LOOK FOR NO SUBSTITUTES.

use a test light and check at the ligh bulb socket to see if you have power there. also check for ground
this light circuit is connect to your tail lights so check to see if it has an open circuit.

Reverse lights are activated by a small switch on the transmission. Usually located on the passenger side of the vehicle with a small rubber cap over the switch, Your plate light bulbs are probably in need of replacement, and have nothing to do with the reverse lights.

Hey, my 1998 Jeep Grand Cherokee did the same thing and it took me a long time to find a solution. It turned out, at least in my case, that there was a burnt out bulb just above the license plate. When I replaced it the warning went away.

You can find instructions for doing this in any half decent manual. One thing they usually don't tell you is when removing the large axle nut on the end, it helps to put a spare tire on with no center cap and with the vehicle on the ground, break that nut free before starting the job. without air tools, it's next to impossible to remove it when jacked up.

They are on the same electrical circuit as your marker lights. You could have a bad ground at the license plate area.