Loud clicking sound, seems to be coming from alternator. I can physically see the alternator shaking when the noise is louder. If I turn on the A/C the noise gets louder and more consistent and the alternator vibrates like hell. Noise almost goes away entirely when idling in park. As soon as I put it in gear it gets louder as well.

How is the sound when the A/C compressor is off? Before you begin changing parts, check the idler pulley. They do tend to go out and a rattle is the first sign. If the bearing goes out completely the belt will fly off and you'll be stuck. Whatever problem you have, change the idler pulley while you are doing the other repair. It's only about twenty bucks and takes all of three minutes to change.

The clicking sound basically goes away when in park with the A/C off. Driving it is persistent even with A/C off, although it gets louder and when the A/C is on.

Not being able to see that far I suspect the clutch on the a/c compressor. Look at the face of the compressor, the part the belt goes around. The clutch is there. Sometimes you can visually see it as damaged/worn out. Start the vehicle and leave in park. Have someone cycle the ac switch from on to off while you watch the compressor clutch area. YOu should get one click when it comes on and that is it. If the clutch on the compressor shakes, rattles, or vibrates/clatters there is your problem.

The vibration you feel/see sometimes happens when the clutch/compressor seizes up. The belt attempts to turn the compressor but resistance is met placing a drag on the engine.