In general, non-immigrant aliens are forbidden to possess any firearms or ammunition. But there is a big exception for a legal alien who:
...is in possession of a hunting license or permit lawfully issued in the United States
(See Title 18, USC Chapter 44, Section 922, part (y)(2) for details.)
Green-card holders and immigrant aliens who do not yet have their green card are both okay under federal law, although many people (including gun dealers, law enforcement officers, etc.) are not aware of the distinction or the hunting license exception, and erroneously think that either you have a green card, or you can't have guns.
Also, non-citizen cannot purchase a firearm from a federally licensed firerams dealer unless they have been a resident of their state for 90 days. So, no, tourists on short visits cannot walk into guns stores to get "protection" during their trip. Sorry. (I've been asked that multiple times.)