It depends. Did you not even bother to read your own link?Nope, aliens can own firearms.
May aliens legally in the United States purchase firearms? | Bureau of Alcohol, Tobacco, Firearms and Explosives
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)...www.atf.gov