Check your belt and tensioner. If your belt is glazed or the tensioner is weak, there can be enough slippage to allow a wet belt to slip around the A/C pulley without turning the compressor. The car wash is spraying hot soapy water in the grill and possibly from underneath to wash the underside. As soon as you drive away the belt will sling the water off and once again gain traction on the pulleys. You will see this most often with the compressor, alternator or water pump pulley.
As a test, bring the Jeep up to operating temp with the A/C running. Pop the hood and spray a lite mixture of water and dish washing liquid onto both sides of the belt. You don't need a lot of soap. Maybe 10:1 or so. The kind of mix you would use to locate an air leak on a tire. Watch to see if the compressor stops turning. Could be the belt slipping, could also be a wet clutch. Now spray clean water on the belt to rinse away the soap. Don't spray the engine, just rinse the belt. As the belt slings the water off, you will see the compressor start turning again.
If this test is positive, it does not mean the belt or tensioner need replacing now. Determine the health of the belt or tensioner seperately from the soap test.
But... it does happen (more) when they are worn than with a tight new belt. And it might happen in a thunderstorm or water puddle splash. Long belts don't like water. Flat belts can hydroplane just like a tire.
Older Dodge Caravans were bad about the belt popping off in the rain. The solution was an aftermarket kit with grooved idler pulleys and a belt with grooves on both sides.
Another thing to remember is this... metal expands when it gets hot. But a rubber belt contracts when it gets hot. So if the belt gets sprayed with water, the temp of the belt will drop fast. Thus it gets a little longer and less traction. And the tensioner gets a little less wound up too. A few fractions of an inch is enough to make a difference. Ever drive an old car where a cold V-belt squeals until the engine warms up?
Check your belts and tensioner health.