I'm afraid there probably isn't one.
Growing up with some Native family myself, I gathered "Happy Hunting Ground" was not an actual native term they used, but rather one of those phrases white men use to make fun of them, like "firewater" and "thunder-stick". That's the context I always heard it in.
However, there appears to be a legit Wikipedia page for it, so its gotta have been A ThingTM, right? And that page mentions the Lakota by name. Those people are closely related to my own Osage, with very similar names for what are usually pretty much identical beliefs. That's significant, because there exists an Osage/English dictionary, created by one of the world's first indigenous linguists way back in the late 1800's. This was when the Osages still remembered their old beliefs, and that dictionary is chock full of religious and ceremonial terms. If it exists, it must be in there. I happen to have a copy, and....
Short answer: that's not in there. Nothing remotely like it is in there.
So I think, perhaps this is my first ever encounter with a large religious difference between Siouxan peoples? The online Lakota dictionary must have it...
Nope. Nothing like that under "happy" or "hunting." In fact, if you try looking up "heaven" there, you get the following:
mahpiya wokichunze n heaven (Christian concept; coined by the missionaries)
In other words, the Lakota didn't have anything like that. The above two words are actually the Lakota words for "sky" and "kingdom", slammed together.
A bit more online ...er... hunting, netted me this explanation for the mystery:
Concerning beliefs regarding an afterlife among Plains Indians, Sioux
physician Charles Eastman writes: “The idea of a ‘happy
hunting-ground’ is modern and probably borrowed, or invented by the
Charles Eastman wasn't just a physician, but an acknowledged lecturing expert in Sioux history. Born in 1858, his maternal grandfather was a Santee Dakota chief, and his Dakota name was Ohíye S'a. We aren't going to get much closer to a primary source.
So the "Happy Hunting Ground" is indeed a myth, but its a white man's myth
OK then, you might ask, what was the native view of the afterlife?
Again, nothing's universal. However, the Siouan people largely shared an animist belief in a omnipresent spirit they all call something like "Wakonda". This is often translated as (more literally) "Great Mystery" or (more commonly) "Great Spirit". Europeans found it easiest to equate it to God (or The Holy Spirit), but the best modern analogy is probably to The Force, the Jedi religion from Star Wars. It is not personified in any way, has no gender, and it permeates everything and everyone. From this point of view, when you die, your spirit doesn't really go anywhere. It just becomes one with Wakonda. It might even be available for consultation from time to time (say if a Death Star needs blowing up or something).
I understand some unrelated tribes had similar beliefs, but this almost certainly wasn't universal of course. The Lakota, Commanche, and Apache languages (and thus the roots of their cultures) are as different as Han Chinese, Turkish, and Benghali are in Asia.