Background check is a requirement for regular gun sales as well, except private person-to-person sales - in practice, this is the vast majority of transfers.
ATF can be more thorough with NFA items because the law doesn't have a limit on how long they can look at you, unlike those regular NICS checks which have a hard limit - but the list of things that makes one ineligible to own is the same.
As far as residency, you have to be a resident somewhere in US, but you don't need to be a permanent resident / green card. A student or a work visa is good enough, combined with proof of current residency (such as utility bill with your name and address).
This isn't quite what OP asked for, of course - you can't have one "shipped to your doorstep" - but this is also true for most regular firearms (there's a collector license that enables this for some old guns).
ATF can be more thorough with NFA items because the law doesn't have a limit on how long they can look at you, unlike those regular NICS checks which have a hard limit - but the list of things that makes one ineligible to own is the same.
As far as residency, you have to be a resident somewhere in US, but you don't need to be a permanent resident / green card. A student or a work visa is good enough, combined with proof of current residency (such as utility bill with your name and address).
This isn't quite what OP asked for, of course - you can't have one "shipped to your doorstep" - but this is also true for most regular firearms (there's a collector license that enables this for some old guns).