It's kinda iffy. By the letter of the law non-immigrant aliens that are in the country under 180 days cannot possess a firearm or ammunition unless they came here specifically for hunting or sporting purposes. How do you define "possess"? Well, in the case of a felon the Gov defines touching as possession. . .
Look in Title 18, USC Chapter 44, Section 922 (y)(2). There's a lot to it, but it boils down to if you are in possession of a hunting license or permit, you're good to go. Which you *can* get on a tourist basis. You're also good if you are a foreign official or have a certification in your home country or get approval from the state department.
To the letter of the law, no, he can't shoot.
But I've taken foreigners to ranges before, private and government run. It depends on who is running the counter and what the specific store policy is, and the overall appearance of the foreigner. If they look respectable, you probably won't get much guff.
Excerpt, emphasis added.
(2) Exceptions.--Subsections (d)(5)(B), (g)(5)(B), and
(s)(3)(B)(v)(II) do not apply to any alien who has been lawfully
admitted to the United States under a nonimmigrant visa, if that
alien 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;
(B) an official representative of a foreign government who
is--
(i) accredited to the United States Government or the
Government's mission to an international organization having
its headquarters in the United States; or
(ii) en route to or from another country to which that
alien is accredited;
(C) an official of a foreign government or a distinguished
foreign visitor who has been so designated by the Department of
State; or
(D) a foreign law enforcement officer of a friendly foreign
government entering the United States on official law
enforcement business.
(3) Waiver.--
(A) Conditions for waiver.--Any individual who has been
admitted to the United States under a nonimmigrant visa may
receive a waiver from the requirements of subsection (g)(5),
if--
(i) the individual submits to the Attorney General a
petition that meets the requirements of subparagraph (C);
and
(ii) the Attorney General approves the petition.
(B) Petition.--Each petition under subparagraph (B) shall--
(i) demonstrate that the petitioner has resided in the
United States for a continuous period of not less than 180
days before the date on which the petition is submitted
under this paragraph; and
(ii) include a written statement from the embassy or
consulate of the petitioner, authorizing the petitioner to
acquire a firearm or ammunition and certifying that the
alien would not, absent the application of subsection
(g)(5)(B), otherwise be prohibited from such acquisition
under subsection (g).
(C) Approval of petition.--The Attorney General shall
approve a petition submitted in accordance with this paragraph,
if the Attorney General determines that waiving the requirements
of subsection (g)(5)(B) with respect to the petitioner--
(i) would be in the interests of justice; and
(ii) would not jeopardize the public safety.