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There two wires to the A/C compressor clutch, the Blk/Wht wire supplies power from the A/C Fuse and the Grn wire goes to the Blower Control Module (BCM).
When the BCM calls for A/C it grounds the Grn wire to pull in the clutch.
First check the A/C Fuse.
You should see the voltage on the Grn wire go from +12Vdc to zero when you turn on the A/C.
If the BCM isn't bringing in the A/C Clutch you can test the clutch operation by tapping a ground jumper against the Grn wire from the clutch, it should pull in. BTW, this is wire color coding for an '88 Vette with Electronic A/C. With manual A/C the color coding at the A/C Compressor Clutch differs.