There is no Class III license to POSSESS a full auto weapon- it is to SELL full auto weapons. When purchasing one, the transfer is THRU a Class III dealer.
If you locate a dealer you want to do business with, AND if your STATE allows private citizens to possess full autos, AND you have a balance in your checkbook that will cover the EXTREMELY high prices for most full autos- your dealer will guide you thru the paperwork. It requires a background investigation, fingerprinting, approval by your local chief law enforcement officer, and the payment of a $200 transfer tax. Your state may ALSO have registration requirements.
This is a real quick and dirty, but bottom line, find yourself a Class III dealer.