Here is a link to fixing the blend door motor. There is a photo of the motor & other good info.
Blend door motor repair
Here is a link to the wiring diagram. It is for a little earlier model & some of the wire colors are different.
What has to happen is the top wire on the 3 wire connector (blue/orange on yours, I think) on the mode switch has to connect to the bottom (black) one when on a/c or defrost. That should be the first 2 & last 2 positions on the dial. That grounds the relay to start the compressor. That assumes the high & low pressure switches are closed. If the pressure gets too high or low they are to open & stop the compressor to protect the system. One of those switches could be bad & open the circuit preventing the compressor from engaging.
Here is a link to some FSMs. They don’t go past ’03 but in the a/c chapter of the ’03 in the controls section there is info on the pressure switches & the blend door motor that will apply to yours. The a/c wiring diagrams are in the wiring chapter as well.
See online read only FSMs here:
Jeep Knowledge Base
You may have to go to the bottom of the page & pick Jeep Resources & online manuals if it directs you to home page rather than the knowledge base.
Pick your model & year then on the drop down menu pick the chapter you need.