"Non-U.S. citizens that are in the country legally can have a PFEC conducted. You must include your Alien Registration or I-94 number on the PFEC application, and include a copy of your Alien Registration or I-94 card with your application package. A valid Alien Registration or I-94 card is required at the time of purchase of any firearm. Non-U.S. citizen applicants must also meet certain federal requirements to purchase a firearm, unrelated to the firearms eligibility background check. "
PFEC = Personal Firearms Eligibility Check
Go
here for all the forms and regulations, etc.