You can build the closed bolt, semi auto STEN's from CATCO. These are legal. ATF won't approve an open bolt semi because of easy convertability.
If the entity is incorporated, it is a corporation (LLC/LLP are primarily tax issues). Corps. are not required to get LEO sign off.
Periodically, pre '86 STEN receivers show up. If you had one, or could get one, you could build the gun. The receiver is the "machine gun" and in this case would have existed before the ban and building would be legal.