In general, the right to keep in bear arms exists for all people in the United States, not simply citizens, so at a federal level, there is no bar (provided you enter the country legally -- federal law bars people in the USA illegally from possessing firearms.)
Some people have mentioned that state laws concerning ownership and possession of firearms vary greatly. This is true, but those laws tend to impact the ease by which citizens or resident aliens might go about buying, possessing, or carrying those firearms for self-defense purposes, or using them for hunting or other sporting purposes. For someone who simply wants to come in to the country, go to a range, and shoot a handgun or rifle for an hour under supervision, there generally won't be much of an issue.
I would be more concerned with availability of ranges. In some cities in some locations, there are few gun ranges within the city limits. New York City or Chicago are classic example, so (oddly) Detroit. Generally cities in the South, Midwest, or Rocky Mountains will likely have more such facilities. (As does Las Vegas). In general, it is my experience that even if you go to a large city in a state with more restrictive gun laws (like, say, California) there will be ranges who would be willing to cater to your desires. (It is undisputable in American jurisprudence that the Constitution protects an individual right of the "people" (not just citizens) to keep and bear arms, and while states may attach various regulations on that right, they cannot stop people from possessing firearms.
A lot of times, gun ranges are mentioned/reviewed on Yelp. I encourage you to call a few people in the place of your final destination in America and ask some questions.
As with any activity involving things that go "bang" don't forget that your first goal is to come home safely. Pay attention to all range safety rules, and especially the "four rules" https://www.itstactical.com/warcom/firearms/colonel-jeff-cooper-carry-conditions-and-firearm-safety/