Do research grants from USA governmental agencies (e.g. NSF, NIH, DOE, etc) require that all of the PIs be affliated with a university?

or to other research organizations for example government labs?

I am currently a post doc at large and would like to co-write a proposal (probably NSF) with a Professor at a nearby University. With the funding, I'd be getting a salary and doing the majority of the work and the lead PI (the prof) would work partial time and time in the summer.