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Imagine an ordinary foreigner, arrived with a simple tourist visa into the US.
In the U.S., there is no personal identification document. Most of the U.S. people doesn't have a passport.
What prevents a simple tourist to go into a weapon shop and buy a rifle?
18 USC 922(g) says that it is illegal for various persons
to ship or transport in interstate or foreign commerce, or possess in or affecting commerce, any firearm or ammunition; or to receive any firearm or ammunition which has been shipped or transported in interstate or foreign commerce.
(5) refers to "being an alien", specifically one who
has been admitted to the United States under a nonimmigrant visa (as that term is defined in section 101(a)(26) of the Immigration and Nationality Act (8 U.S.C. 1101(a)(26)))
with an exception
as provided in subsection (y)(2),
The (y)(2) exceptions covers foreign officials and law enforcement (not a typical tourist), and one who is
(A) admitted to the United States for lawful hunting or sporting purposes or is in possession of a hunting license or permit lawfully issued in the United States;
which pretty much covers simple tourists, if not typical tourists.
In the U.S., there is no personal identification document.
That's not really true. There is no standard personal identification document issued by the national government (if you don't count passports), but state governments do issue such documents. The most common is a state-issued driver license, and people who do not drive can be issued a "non-driver ID" that is otherwise similar. Practically every US citizen or resident has one of these documents, and in practice they are used in a similar way to a national ID card. (You could do an image search to see what they look like.)
In fact, there is an identification requirement for buying firearms; see https://www.atf.gov/firearms/qa/what-form-identification-must-licensee-obtain-transferee-firearm. A state driver license or non-driver ID would be sufficient for this purpose, and that's what most people would use.
The link mentions that a foreign passport would be acceptable identification; but that doesn't mean a foreign tourist could actually buy a gun. The answer by user6726 cites the relevant law that restricts nonimmigrant aliens from firearm purchases.
