Do research grants from USA governmental agencies (e.g. NSF, NIH, DOE, etc) require that all of the PIs be affliated with a university?
Every US funding agency has its own rules, and most agencies have many different types of programs with different intentions.  For most (maybe all) agencies, this includes both funding to universities and funding to corporations, particularly under SBIR programs.  Some agencies will also fund individual consultants directly for various specialized roles, but that tends to happen mostly with rather senior individuals.  

Your options are even larger if you are not the PI, however, but a co-PI (I think this is actually what you want to do).  For most agencies and most funding vehicles, the PI's organization can subcontract work with a fair degree of freedom.  Also, there is only one PI, from the point of view of the contract: it doesn't matter if you call all the leaders PIs, the government is going to negotiate with one organization per contract, and that organization is going to be asked to designate one person be the PI, and in practice everybody else is co-PIs.

For you, then, there are three main paths to take:


Propose, with the professor as PI, under the plan that if funded you will become affiliated with the PI's university
Propose, with the professor as PI, where you will be contracted as a consultant.
Set up an LLC, from which you can propose as PI, and subcontract to the professor.  You'll have to comply with government contracting rules, which are painful, but much less so for SBIRs.


Number 1 is probably the best, unless you have some reason to avoid affiliation, in which case #2 is a good alternative.  #3 doesn't have much advantage unless you intend to apply for SBIRs, and is still a pain to get started.