You'll find Indian Reservation hunts in several states, and you do not need any license other than a tribal license. As for elk hunts, in Arizona you have White Mtn., San Carlos, Hualapai, Havasupai and Navaho that all offer some level of elk hunting opportunity. As you can imagine, they're generally very expensive with waiting lists, but if you do your homework, there are some neat hunts that aren't as expensive. One hunt I'd love to do if my son were still young enough is the youth elk hunt on San Carlos. They limit the size of bull a kid can shoot, but it's only $3,500 or so. In New Mexico, you might consider the Mescalero, Jicarilla, Pueblo of Acoma or Laguna Pueblo. The Jicarilla has traditionally offered some of the less expensive elk hunts on Apache reservations. I hunted on Laguna about 5 or 6 years ago, and had a bad hunt, but have heard that they've made changes and might be a better option now.
You can find Indian Reservation hunts in most other states. I've also hunted whitetails in Montana on a reservation and had good results up there. You do have to keep in mind that each reservation hunt will be different depending on the tribe and what you're hunting. I've found hunts as inexpensive as a few thousand dollars all the way up to $40K for some of the elk offerings on San Carlos.
By the way, all the indians I know call themselves INDIANS, not NATIVE AMERICANS. I have a buddy who's a Cree that guides me when I hunt whitetails on his rez in Montana. He told me that the term Native American is for white folks who need to feel better about themselves; and Indians are proud to be referred to as just that, INDIANS. He told me they'd rather be referred to as Apaches, Navajo, Cree, Shoshone, Blackfoot, Utes or whatever, than Native Americans.
