What is causing the car to automatically shut off as soon as I push the AC button?

could be that the fast idle stapper motor valve is not working which provides more air to work with ac power usage if that is the cause try cleaning your throttle body by a cleaner to clean its holes

first thing to check is the AC compressor, some models you can turn the pulley (clutch assy inside the belt pulley) by hand when the engine is stopped, if not take the fan belt off that runs your AC compressor turn the ignition to run position turn on your AC (button and fan speed on high) then try to turn the compressor pulley by hand if you can not move it then its seized and requires a replacement unit. That's what it sounds like to me had this happen a few years ago on MK2 golf but instead of stalling the engine it spun the belt off as soon as the compressor was engaged. GL