A quick google suggests that if you want your tank to have working guns and/or turret you need a Federal Destructive Device Permit, which includes a background check and ATF approval.
Destructive devices are NFA items, true. If you want to own a DD, you need to pay the $200 federal transfer tax, which is done by submitting a form to the ATF and getting a tax stamp from them.
It's not a "permit", though. And there are no special limits on who can own one - if you can legally own a gun, you can legally own a DD or any other NFA item. One doesn't even need to be a US citizen or a permanent resident for that, even people on student and work visas can do it.
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).
