The actuator/door that controls recirculation constantly ticks, whats the part number-how to swap?

I can see the confusion. You're looking for the part with callout 87106N in this drawing, which is part number 87106-47120 according to the online catalog, but the names are messed up. It's the inlet door actuator, but the catalog has that name on the 87106J part instead, which is the outlet select actuator (I think). So somebody needs to flip a coin over whether the online catalog is wrong about the part numbers or wrong about the names.

On the other hand, you probably don't even need a new one. The way these generally go is the position sensor gets dirty and gives an erratic position signal. The ticking/wiggling sound is the ECU sending the door back and forth in a futile effort to get the erratic position signal to hold still.

Once you get the actuator out, you can most likely just clean the sensor part, as described in this thread. The only bad news is this is the hardest actuator to get to. Cleaning it up won't take you ten minutes once you get there, but getting there will be a job and a half.

One thread where I remember somebody wasn't able to just clean up the sensor tracks was where they had lived with the problem so long that the incessant waggling had worn the sensor traces all the way through. So a new actuator was needed in that case.

 

https://attachments.priuschat.com/attachment-files/2018/05/146483_AC.pdf
Page AC 30 for schematic

A8-22 and A8-23

12 volts to A8-23 P - W-B
Neg to A8-22 G - W-B
Sets it to Fresh

Reverse that for Circulate.

Set climate to one of the other, turn car off, then pull the A8 plug and use a battery to move it back and forth a lot to SWIPE the wiper and see if it cleans / clears it

When done, make sure to SET it back to what you set on the car climate control

The car is OFF while you do all of this.

 

The Gen 2 Prius has various potentiometer-based sensors that work this way: all three HVAC servos and the coolant control valve, at least. Earlier, in Gen 1, you could add the go pedal and the power steering torque sensor as well.

They have all had known problems when the contract traces get electrically noisy with age, and generally people who have tried disassembly and cleaning with electrical contact cleaner have had good results, but just trying to mechanically swipe the wiper back and forth has been ineffective. After all, that's what happens in normal operation anyway, and the traces still get noisy.

The ultimate solution to those problems has been for Toyota to get rid of that kind of sensor. By Gen 2, they had already replaced the steering torque and go pedal sensors with non-contact magnetic or Hall effect sensors. By Gen 3, they had also gotten rid of the potentiometer HVAC servos, and used sensors that make position pulses instead.

I'm not sure there are any potentiometers left as of Gen 3, aside from maybe the panel light dimmer and the windshield wiper delay.