ATF website:

An alien legally in the U.S. is not prohibited from purchasing firearms unless the alien is admitted into the U.S. under a nonimmigrant visa and does not meet one of the exceptions as provided in 18 U.S.C. 922(y)(2), such as possession of a valid hunting license or permit.

[18 U.S.C. 922 (d)(5), (g)(5) and (y)(2); 27 CFR 478.11 and 478.32(a)(5)

I was a bit surprised myself! Now to let my wife know she can buy me a gun for Christmas!