If the rear a/c runs on its own switch, check the switch, and if it works, you can check the blower motor, as the rear a/c has its own blower motor separate from the front. Should be located within the housing of the rear a/c. If the vehicle is still covered under warranty, this problem may be covered.
Attached to the blower motor under the passenger dash
The bearings are out of it. Replace Blower motor.
switch contact"s
The blower motor is drawing to many amps, replace the blower motor.
You either have a blown fuse, defective blower motor, defective blower motor switch, or a defective blower motor resistor pack.
blower resistor is bad
The blower motor resistor is located under the glove compartment in a black shroud.
Blown fuse.
Replacing the 1997 Chevrolet suburban blower motor resistor can correct your air output issue. The air output issue could because by other things.
With no speeds at all, check the fuse, then check the blower speed switch and it's electrical plug.
It is usually under the dash on the passenger side right next to the blower motor. On some, it is mounted on the same bracket as the blower motor.
Wright above the passengers feet under dash.