For big game hunts in the USA there are basically 2 types. 1) purchase a game license from the state you are going to hunt in and obey / go by the rules of the state game department. Big game hunting is seasonal so the hunts are for specific days. Often the “days” are spelled out and one week or as much as 3 weeks in duration.
The game license for this would be an “out of state license”. IE you are not a state resident. For some species (Elk) the permit tags are limited and there is a lottery OR first come first served. Some folks sign up every year but only get one tag every few years.
You can hunt on your own, or use the services of a guide -your choice. My understanding is that Alaska is a special case and does require a guide for out of state hunters for some species.
Big game hunting is possible in all of our 50 states. Only some of the states have the game species you are interested in.
Hunting rifles or ammo are no problem here. Not sure about US customs entry declarations needed.
The 2nd) type of hunt is on private ground that is set up as a hunting presurve. These are fenced area's that stock their own animals and can have their own hunting rules. Some of these “hunting areas” are small, less than a square mile, and some are much larger, several square miles.
Some links that may be useful
this is safari club
this link is for Colorado state game department
this is for the state of Montana