Speaking as a Permanent Resident Alien, yes, you enjoy all the rights of a Naturalized Citizen.
On a federal level, when purchasing a firearm, I have to provide my A# (My 'Green Card' ID) but aside from that, it's the same as an American Citizen completing a 4473. I can, and have purchased NFA items. Without an A# (and hence, that "permanent" status) I don't think the ATF will process the 4473
At the state level, an Alien may be required to provide more evidence that indicates that your residence here really is permanent (and that'll vary state by state) but in Connecticut and Pennsylvania, beyond a utility bill, everything was the same.
I guess it's possible that the background check may be a little more extensive, but I've never had to provide other documentation.