The latest news on this front comes from a really interesting article at Ballot Access News written by the widely-respected election expert and advocate for third party/independent ballot access Richard Winger:
On May 14, Rhonda Brown Fleming sued the Cherokee Nation in U.S. District Court in Washington, D.C., over whether she should be allowed on the ballot for the June 1 election for President of the Cherokee Nation. She is a descendant of a slave. Until 1866, the Cherokee Nation permitted slavery.
She was kept off the ballot because she is not a blood member of the tribe, but she argues that it had previously been determined that descendants of slaves held by members of the Cherokee Nation are also part of the tribe. Here is the complaint. The case is Brown Fleming v Cherokee Nation, 1:19cv-1397. The case is assigned to U.S. District Court Judge Thomas Hogan. She was also kept off the ballot because she does not live on the reservation.
The Cherokee Nation organize their own elections separate from state or local law as explained by Your Vote Your Voice on their Tribal Constitutions and Voting webpage:
Approximately 60% of today’s tribal constitutions were adopted under the Indian Reorganization Act of 1934 . The constitutions set forth the framework of tribal governments, for example the role of a tribal chairperson — sometimes known as a chief, chairman, or a president — and whether they are selected by election or heredity .
Tribal constitutions also outline tribal voting procedure such as when elections occur, who is eligible to vote, where polling places will be, what time polls will open and close and the specifics of absentee voting . Tribal constitutions generally weave together traditional forms of government alongside a contemporary government framework with executive, legislative and judicial branches.