Is it possible for aliens to introduce their own firearms to hunt in U.S.?

Overall, non-immigrants aliens may not introduce or posses firearms in U.S. territory unless they meet requirements for an ATF exception. On this regards, aliens with a valid hunting license or permit are among those allowed to temporarily import their own firearms to U.S. as well those admitted for lawful hunting or sporting purpose.

According to the aforementioned, it's possible for an alien to introduce its own firearm if he/she has a valid hunting permit.

In order to comply with regulations it's necessary to have an approved ATF Form 6 NIA, which is valid for one year after the approval date. The ATF Form 6 NIA must be shown to the Customs and Borders Patrol Officer with the proper documentation demonstrating that you are elegible for an ATM exception regarding temporarily firearms importing to the U.S.

So, a valid hunting license of any of U.S. state is required as well any other particular document depending of your condition.

All firearms temporarily importe to the U.S. under an ATF Form 6 NIA must be removed from the territory at the end of the owner's visit. Such firearms can't be sold, rented or any form negotiated within the U.S. territory, then the owner only has permit to introduce the firearm for the declared purposes and must be committed to  take it away at the end of the visit.

It's important to be aware that not all types of firearms and calibers are eligible for a ATF Form 6 NIA. Refer to the ATF website for further information about your own firearm eligibility for an ATF Form 6 NIA. 

Take in mind that you might need special permits besides the ATF Form 6 NIA, especially if you are planing to arrive to the U.S. by plane. On this case consult with the airline staff to know special permission to transport firearms. 

Furthermore, if your flight makes stops on any country different from U.S. after leaving your homeland, you need to be aware of transit countries regulations regarding firearms transportation.

Remember that once your travel finishs, all firearms as well ammunition must be removed from U.S. territory; so be aware also about your own country regulations about firearms and ammo repatriation.

Finally, if all this paperwork is complicated for you or if you simply don't wish to deal with tons of permits, it's possible for an alien to rent a firearm in U.S. for hunting purposes.

Requirements for renting a hunting firearm are the same that those needed for a temporal import form. The only difference is that you won't carry the firearm with you.

Then, in order to rent a hunting firearm you need to have a valid non-immigrant Visa as well a valid Hunting License issued by any state.

Remember that in both cases, temporal import or rent, you must comply with Federal and State regulations regarding firearms use and transportation, so be sure to be aware of all the laws you need to comply with in order to avoid unnecessary problems while using/transporting your own or rented firearm in U.S.