The low pressure cutout switch will shut power off to the compressor when the refrigerant gets low.
If the compressor won’t run with the A/C switch in the A/C on position, then the system is likely to be low on refrigerant. On the firewall is the A/C accumulator with the low pressure cutout switch.
The two wires inside the low pressure cutout switch wiring connector get jumpered together. That bypasses the switch and turns on the compressor when the refrigerant level is low. Don't run the system very long, because low refrigerant also means low oil. The refrigerant causes the oil to circulate though the system and lubricate the compressor.
The low pressure cutout switches are adjustable. They should be set with the low limit about 18-21 PSI if the system has been properly converted to R134a, the stock setting is about 25 PSI. Remove the wiring connector and there is a screw slot in between the connector pins. Turning it clockwise increases the trip pressure where it cuts off and counterclockwise decreases the trip pressure. I believe that each 1/4 turn counterclockwise lowers the cutoff pressure by 1 PSI.