Absolutely. For that matter, I could pay armed guards to patrol my personal property as a private citizen - although generally this would be a like a celebrity with armed body guard(s) situation where they were actually employed by another entity who sold their services to me. See below why the average private citizen probably doesn't have the know-how or manpower to maintain a person security force of their own (which may fall under what you call "government involvement" - not sure what you mean by that).
Any weapon that is legal to own and operate in the United States should be legal to arm a private security force with. However, per the law, the employer must demonstrate they are maintaining some kind of control over their armory, are limited as to what and how much they can store and where, and generally fall under the authority of the Bureau of Alcohol, Tobacco and Firearms (ATF), although they may also fall under the inspection authority of local law enforcment and/or fire marshalls and the like. Additionally, private security guards generally have to be licensed/certified, generally at the state level, even to provide unarmed security services. Obviously arming them requires further certifications/trainings/background checks/etc. In those senses, there is "government involvement."
If I just get a bunch of my boys together, we take over an apartment block and we start armed patrols of the neighborhood? We are a gang - no government involvement.
If I get a bunch of my boys together, we move onto a compound in the middle of the desert and start armed patrols of it? We are a militia - no government involvement.
Obviously not the same thing as an armed private security force.
Government involvement, I meant if the private company or individual is manufacturing or researching on the government’s behalf.
Say Tesla got a contract to make solar powered tanks. I would guess that the government would either supply security or allow them license to have whatever reasonable security measures in place.
Yes I gathered that any staff would have to be licensed/permitted at a state level.
As you say, getting a bunch of dudes with guns together is not strictly legal, depending on the circumstances. Gun club, fine, vigilante group, not fine.
But the long and the short of it, provided one seeks appropriate state, federal and any other required licensing, they could have employees wandering around a private business, openly carrying fully automatic weapons, ready to have a shootout with the A Team or Gordon freeman, if the need arose.
One thing I wondered, but didn’t ask was that in these circumstances, can a CEO seek sheriff status, and deputize employees as they see fit, or would the office of sheriff mean they would be responsible for law enforcement in the administration and could not be used as a way of employing security in a company with full powers similar to police officers?
I realize that sheriffs are often elected and in some places operate more like a mayor.
So I understand if there is not a simple answer to the question.