During Voyager, Chakotay's Native American heritage is repeatedly mentioned and emphasized. What tribe is he from? Do tribes as such still exist in his time? Does he hold membership in one or more tribes?
Chakotay's tribe and their beliefs were referenced in numerous episodes. Per this Memory Alpha page linked by Paulie_D. (emphasis added)
Chakotay's tribe was a group of Native Americans descended from the ancient Rubber Tree People... Their ancient language, also shared with their cousins in Central America, appeared almost unchanged from that of the Sky Spirits.
The Rubber Tree People were an ancient Native American tribe who lived in the rainforests of Central America on Earth.
Tattoo confirms that the "Sky Spirits" who visited and genetically modified the Rubber Tree People in 43,000 BC were actually Ancient Aliens (TVTropes link!) that, by amazing coincidence, originally came from the delta quadrant.
Members of the tribe lived on a colony (possibly Dorvan V) near the Cardassian border in Chakotay's time, although some remained in the jungles of Central America to live in the way of their ancestors.
Out of universe, the Rubber Tree People are not an actual native tribe, although they were intended to be a "pre-Mayan culture." There was some criticism of the ritualistic practices shown in Voyager, which appeared to be an amalgamation of native stereotypes as written from a European perspective. For example a "Mayan medicine wheel" is shown, which doesn't exist in real life. The actor who portrayed Chakotay, Robert Beltran, is not a Native American. Here's a random essay I just found that discusses the topic in depth.
It is strongly implied, though not explicitly ,that Chakotay's group were followers of some form of Neo-transcendentalism. They were not an old, historic culture... they were members of different ethnicities who had lost their ancestral traditions and revived them, intermixing anything they could get their hands on. Strictly taken they were "fake-natives" and Chakotay seems to have been aware of it as he never took his ancestral tradition quite serious until he was somewhat older and matured and finally got to respect his tribe for what it was and accepted and endorsed the culture they had made on their own.