Found an interesting Smithsonian Folkways article about Hopi songs recorded in 1924. It was an official recording by the Smithsonian Institution and Bureau of American Ethnology so I feel comfortable linking it here. These are Hopi chants / songs and were recorded in Arizona so it is only representative of that tribe.

In the PDF liner notes there are mentions of recordings made as early as 1889, but I couldn't track down any samples. Very interesting read nonetheless.

It's hard to say what general "stereotypical" native american music might consist of, but in my experience Hollywood Westerns do tend to use somewhat authentic music for their depictions of Southwestern US and Plains Indian cultures. I used to attend Ute pow-wows regularly, and the heavy rhythmic drumming, undulating singing, and occasional flute is definitely in line with what you'd hear in a Western.

Of course, whether or not modern pow-wows are a good representation of the historical culture is absolutely up for question and I sadly can't find any real academic information on it. Microstudies of specific Native American cultures are ridiculously hard to come by, at least for a amateur like myself.

More generally, this seems worth a read. The details about two-three tone chanting and AABB or ABAB rhythmic structures in the Paiute section definitely correspond with what you hear today - not surprising however, since this survey is based on recordings made in the 1940s.

You might repost this question under /r/ethnomusicology.

Countless western televison dramas/movies that feature native american themes, the "Shakuhachi" flute preset from the korg M1 comes to mind, which is a japanese flute sound, emulated on a japanese keyboard, manufactured in the 1980s. In many synths and romplers produced in the 1980s and 90s, the sound bank presetd were divided by sound category. And since there is no "Native American" music bank in any rompler/sampler/synthesizer, many sounds came from the "Ethnic" section. Which is really just a mish mash of non-western "world sounds". If you are programming soundtracks in the 80s, you get a couple patches from the ethnic section, and vibey pad sounds and you can just imagine an indian standing infront of a sunset looking down some canyon.

I don't have an expert opinion on this, but anecdotally, I have encountered radio stations in the Navajo and Pine Ridge Lakota reservations playing what might be described as "powwow music", very much like what you would find in a Western. Apparently it's authentic enough to be taken seriously on actual reservations, so I'd guess it's pretty accurate.