No, WRONG!
As usual!
Open your eyes and close your mouth,
(ahem)
All an immigrant needs to legally buy a gun is:
A valid ID issued by resident State
An I-90 Form USICS#
Fill out the Form 4473 and pass the NICS scrutiny
If the immigrant in question is here on a work permit or student visa, and are not permanent residents, in addition to the above, they need to have a valid hunting license issued by the State in which they reside.
Or, they get their girl friend to buy it for them.