If your vehicle is showing engine idles rough or shudders when the a/c turns on, here are the most likely causes and the parts you'll need to fix each one.
The engine computer uses the throttle body or idle air control valve to increase engine speed to compensate for the heavy load of the A/C compressor. If carbon buildup restricts airflow or the valve is failing, the engine RPM drops too low, causing a shudder.
Engine mounts isolate engine vibrations from the cabin. When the A/C compressor engages, it puts a heavy torque load on the engine. If the rubber mounts are torn or collapsed, they will transfer this vibration directly into the chassis, feeling like a severe shudder even if the engine is running fine.
Worn spark plugs or weak ignition coils might perform adequately at a normal idle but fail to deliver a strong enough spark under the increased engine load of the A/C compressor, causing a slight misfire and shudder.
If the A/C system was recently recharged with too much refrigerant (often via DIY cans), it creates excessive head pressure. This forces the compressor to work much harder than designed, severely bogging down the engine when the clutch engages.
Internal wear, lack of refrigerant oil, or catastrophic internal failure can cause the A/C compressor to bind up. When the clutch engages, this extreme mechanical drag pulls the engine RPM down drastically, nearly stalling the vehicle.