Why would air conditioner compressor work while driving but stop working when the car is idling?

Possibly a loose belt. At higher engine speeds, even with a slipping belt, the compressor could be turning fast enough to produce some cooling. But at idle speeds, the loose belt may not turn the compressor pulley at all. == {1}Loose belt {2}Low compressor pressure {weak compressor}

Answer

An undercharging alternator (anything less than 12v) can also lead to the a/c compressor not working while idling, but will work again when the engine speed picks up while driving as the alternator will be producing a much higher voltage.

a) How to determine that the Alternator is undercharging

An easy way to know whether it's undercharging or not is to observe the air-conditioner's blower speed. If the air-conditioner's blower speed picks up in tandem with the engine speed and slows down when in idle mode, then the likely problem is due to an undercharging alternator.

b) How to determine that the Alternator is undercharging

Start the engine - switch on the air-conditioner- switch on all lights- switch on the radio & amplifiers (if u have them)- engage 1st gear (or reverse)-----Now read the voltage with a multimeter to see whether the alternator is producing at least 12V (preferably more than 12V). Needs to be replaced if it's anything less than 12V.

Get the biggest available capacity alternator that you can lay your hands on.