(edit to add: it's most definitely a young red-tailed hawk.)
A nature center probably does qualify to legally possess it. As an individual, you can apply for a permit, but your odds of being granted a permit for the purpose of "it would look cool on my shelf" are essentially zero, and applying might get you in trouble given that you're already illegally in possession. Whoever "collected" it should have had a federal permit; the taxidermist (assuming your neighbor didn't stuff it him/herself) needed a separate permit to stuff it; and your neighbor needed a third type of permit (which it's very unlikely they got) in order to legally acquire it from a taxidermist.
Possession of taxidermied birds by "public scientific and educational institutions" is legal, but only if the bird was "lawfully acquired" in the first place. There's not even any exemption for antiques.
The way the law is written, about the only legal thing you can do with a stuffed bird of prey (other than never taking possession of it in the first place) is hand it over to a wildlife enforcement officer. That might get you and/or your neighbor into some trouble. Might be worth hiring a lawyer to help you minimize those risks.
The relevant statute is the Migratory Bird Treaty Act; relevant federal regulations are at 50 CFR 21.
Buteo jamaicensis is listed as a covered species.
Generally, you need a permit to "take or posess" migratory birds, carcasses or parts thereof, or their eggs or nests. The rules are looser for hunters going after "game birds" like ducks, but for birds of prey like hawks the rules are even a bit stricter than for most birds.
(Additional rules apply to endangered species, but this isn't one.)
Overview of taxidermy rules here:
3-200-8: Migratory Bird Taxidermy | U.S. Fish & Wildlife Service