I don't think there would be a problem for someone outside of the US to come here to hunt with a muzzleloader for whitetailed deer. There are a number of routes that you could take with this. It depends on how much money you want to spend as well as what level of success you are looking for.
You can go on a do -it-yourself hunt in many places on public land, but your success rate might be very low. You would have a wonderful experience, but may not get anything.
Guided hunts. Many states have licensed and registered guides and outfitters that offer deer hunting packages. This would increase your rate of success but some guided hunts aren't cheap.
As far as what is the best area of the US for deer, that is a very subjective question.
One could say that Texas would be the best considering the deer population and that many landowners manage their land for sport hunting. But it would be one of the most expensive hunts going. A good deer hunt could run you thousands of dollars. Also, Texas to the best of my knowledge doesn't have any special muzzleloader seasons. They leave that up to the landowner.Another thing to consider is that some guides offer special muzzleloader hunts for deer. One place that may be of interest in central New York State. You could hire one of their registered guides for a muzzleloader deer hunt for a relatively low rate as compared to some places. This would give you a good chance of getting a deer or two and have a great experience.
On the other end of the spectrum, you could go to a state like Maine or upstate New York. Buy a hunting license over the counter and camp out. You would be on your own but it would only cost a few hundred dollars. On the plus side, both states offer special muzzleloading seasons, so if you ran into other hunters, they would be hunting like you are. It would also be much more challenging.There are many deer hunting operations in Texas that offer hunts over baits or behind a high fence. It may not be as much of a pure hunting experience as you would like.
Another option would be to make friends at this board and see if anyone would invite you along for a deer hunt. That may be another option.
As far as muzzleloaders in the US, I am not sure what the rules are for importation but here in the US for the most part, they are not considered to be a firearm. Even here in Massachusetts where the gun laws are stringent, one doesn't need a license for a muzzleloader. You do need one here for a handgun, modern rifle or modern shotgun. You can go to the Thompson Center Arms website. They have a link to all the fish and game websites for each state. That can give you an idea of what the seasons are for each state.