I'm not intimately familiar with your chassis, but there are two things that I would suspect in your case.

First is the ignition switch. After starting the engine only to have the a/c system not come on, wiggle the ignition switch. If you can make it come on this way, the problem is the switch.

Secondly, inspect the belt and its tension. If the belt slips a rotation sensor sends a signal that drops out the system. It may come on after starting again if it has had enough time to bleed off some high side pressure, decreasing the load thus allowing it to start without making the belt slip.

The other more obscure problem could be a bad clutch or locked up compressor. With engine not running reach down and turn the forward most part of the compressor to ensure that it turns smoothly. If it does, jump 12 volts directly to the clutch with engine off and listen for the clutch to click. If it does not, then the clutch coil is open or the air gap is too wide.

Good luck,