So if he wants to buy a gun, should he be allowed to? According to you he has the right to.
He has the right to own guns as long as he's here legally he's a proper person according to the ATF.
Buying it from a FFL is another story though and it depends on his status as a non-US national.
But yes non US citizens (in some cases) are allowed to buy and own guns in the US.
You can even get a LTCH as a foreign national and legally carry in Indiana.