Unless they've changed in the last couple of years the NFL embraces it. They give the teams a list of what they can and can't do with it. You can call yourself the Pineville Falons and use the falcon logo on shirts, helmets, and jerseys. You can't call your teams the Atlanta Falcons and to the same.
I've gotten a few release letters and official cds of the team artwork over the years. Perfectly clean, professional images in different formats of every version of the logo a team uses.
That said I never had to get permission myself, the teams always brought it all to me.
Spent two minutes trying to find it, all I found was this:
"Many high schools also borrow the popular logos of the NFL’s 32 franchises, although this has not caused any problems. Brian McCarthy, an NFL spokesman, said high school and youth football teams are allowed to freely use NFL insignias. “It is inspirational for young players to play football under the same name as NFL teams,” he said."
That may have all change now that the NFL has
USA Football
So yes, I typed all of that and really did nothing to answer your question.