First do you need a 20 amp receptacle? Not if the appliance is less than 1440 watts. Appliance garage receptacles are not counted as counter plugs.
Most limit switches are rated 10 amps or less and i think there is at least 1 door jamb switch rated at 15 amps. Trouble with these switches is they are not suitable for anything but a hinged door and usually must be installed in the hinge side to ensure the plunger is activated. We require the power to be off when the garage is even partly closed and that cannot be done with a door jamb switch.
A limit switch and small contactor in a proper enclosure is what is needed but some inspectors will let you use a relay inside a 6X6X4.
