[int_19h]

You already have the legal ability to own a rocket launcher - it's not any different from any other "destructive device". The main barrier to ownership is finding someone willing to sell you one, and the price they would likely ask for it. There are rich collectors in US who own tanks (with active turret), artillery etc - mostly older stuff, but still plenty destructive.

[Volundr]

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.

[int_19h]

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.

[Volundr]

This is the site that I was looking at: https://nationalfirearmsact.com/nfa-regulated-items/destruct..., which seems to spell out the background check as a requirement. It also says you need to be a resident of the US. Am I missing something?

[int_19h]

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).