Hello, and thank you for your question.
First, I want to clear something up, because your question is in the FHA Loans section, not the Conventional Loans section. PMI, or private mortgage insurance, applies to non-government loans with less than 20 percent equity or down payment. PMI companies are private insurers, and while lenders usually choose the insurer for you, I see no reason that you would not get to pick your insurer as long as the lender could establish the policy without going through a major hassle. You can always ask. As a loan officer, I worked with several mortgage insurance providers, and if a client had a preference among them I tried to honor it whenever possible.
FHA loans, on the other hand, require MIP, or mortgage insurance premiums. They are paid by borrowers, but the Department of Housing and Urban Development (HUD) administers the program. You don't get a choice with any government loan. The VA backs VA loans, and the USDA backs or funds USDA (rural housing) loans.