Letters that are mention are letterhead letters from Chief Law Enforcement Officer (CLEO) or Military requesting demonstration of specific MGs.
FFL 01 are Dealers. They need to get a SOT 03. They can buy transferrables, premay ( also called pre86) MGs without a letter. Post MG they have to get a letter from the CLEO or military of the specific machinegun. This includes needing a a letter even though the post MG has no requirement of a letter. They can not manufacturer their own MGs since they are a dealer. They can deal with other NFA items like silencer, SBR, SBS, AOW, etc.
FFL07 are Manufacturers. They pretty much have all the rights of a FFL01 but more. They need to get a SOT 02. They can buy transferable, premays, and post no law letter MG (MG that came from FFLs that gave up their SOT or shutdown). They do need letters to buy a post MG from a manufacturer or MGs from other FFLs that have demo letter MG. They can manufacture their own MG and sell them to LEO / MIL / FFLs with letters.
Note SOT means Special Occupational Tax.