yes contact the NFL to see how much you will need to pay them for the rights to use their logo. If I remember correctly they want an arm and a leg for such privilege (I am not being paid for this advice and this only my opinion so don't consider it to be legal advice).
It'd also be more of a trademark issue than copyright. Oh, wait. This is a legal issue. I am not a lawyer and though I watched all 20 years of "Law & Order" on TV, I have never played one myself. I know some lawyers, but I never ask their advice on anything. Unlike the GOP, I have no problem with lawyers with the possible exception of this one guy who cut me off going into a court house one time. But he may have been a judge based on the car he was driving.