hello, Scott;
95% of the time, when an A/C system suddenly stops blowing cold air after a few minutes use it is due to a low refrigerant charge. Your A/C system contains a pressure switch that will shut the power to the compressor cluch off if low side pressure drops below a minimum level. Once your A/C has cooled the interior of the car off and low side pressure starts to drop as a result of less heat being transferred to the evaporator core, system pressure is likely falling just low enough to cause compressor shutdown. The reason for this design is that the compressor relies on refrigerant to circulate oil for lubrication; if the system were to operate without sufficient refrigerant, it would eventually cause compressor failure.
One other possibility is evaporator icing due to a low refrigerant charge; usually the complaint would also include reduced airflow from the vents as well.
I would suggewt taking the car in to your favorite local repair shop to have the system charge level checked, and topped off if necessary. As part of this service they will also check system performance to make sure everything else is operating as designed, and perform a leak test to see if you have any unusual leaks present or if the charge is low due to notmal seepage over several years.
I hope this is helpful; if so, please click my accept button! Thanks!