I live in Page which is surounded by the rez damn near. There are actually a lot of deer and elk in certain units on the rez. They are hard to find and hard to hunt, but there are animals on the rez that will make your heart stop for a second. They usually have tags at the begining of the year that they sell to us white folks. I'm pretty sure that you have to be present to purchase them. Then they have a normal draw like AZGFD for their specific units and species. For these hunts you must be navajo or married to one to put in. Here is a link to their site. They actually have a pretty user friendly website, but every thing is either done by mail or in person. In person is the way to go though. You have to go to Window Rock to speak with someone face to face. If I didn't draw a state side AZ bull tag this year I'd be putting in for the Rez for sure.
If you have any specific questions, I'd be happy to answer.