I wanted to provide an update for this as I have found the solution. Hopefully if anyone else is having this trouble this post should help in the future. I have a 2004 4Runner Sport Edition and the temperature gauge stopped working properly after I did an LED replacement from the factory orange to white. The gauge was stuttering after I reassembled the cluster and I narrowed it down to the stepper motor that powers the needle. I discovered that the proper identification for the motor is "6S".
According to a lot of google searching, This motor was used by Toyota and Lexus from 2003-2013. It will work in any position – speedometer (mph), tachometer (rpm), fuel, or temperature. In order to replace it there may be some soldering required that I will write up how to do once I receive the motor.
A simple google search of "Toyota 6S Stepper Motor" will get you the motor you need. I found it as of December 2020 for $20, a fraction of the cost a dealership would charge.
I found a shop that is charging $175 to fix the instrument cluster. My suggestion if your soldering skills are sub-par is to buy the part and take it to a shop that specialized in instrument cluster repair.
Another option is to buy a cheap instrument cluster on eBay, (I bought one for a Chrysler 300 for $40), and practice on that before doing the real deal.