There is no "class III" license for owning a machine gun, or any other NFA item. All it requires that you fill out the proper forms, send in the required photos, finger print cards, and have the CLEO signature, wait for the background check, and give the government $200 for the tax stamp.
If the local CLEO won't sign off in your area, and the NFA item you are seeking is legal in your state, you can bypass the CLEO if you own your own business, or I think you can even do it with a revocable trust.
If you are a dealer and want to handle Class III/NFA items, then you need a Class III FFL SOT, which is the license. This not needed to own one as an individual.
Cyanide