Hi Matthew,
Nothing appearing in the drop down relates to this issue with PyPi: Thonny Search of PyPi no longer working - #2 by Pixmusix
Follow the Pico instructions and save the files in the same directory you have the main.py script.
Here are those instructions: PiicoDev RFID Module - Guide for Raspberry Pi Pico - Tutorial Australia
Please let us know if you hit any snags with this process.