American Indian law is intensely complicated, but the first thing to state is that non-citizens and foreign governments can bring cases in U.S. federal court as long as at least one party in the case is a citizen. That particular statute post-dates the Indian Citizenship Act of 1924, so I'll continue working backward.

The second thing to bring into the picture is the 14th Amendment and Reconstruction-era Civil Rights acts, one of which, passed in 1870, (42 USC 1981) states:

All persons within the jurisdiction of the United States shall have the same right in every State and Territory to make and enforce contracts, to sue, be parties, give evidence, and to the full and equal benefit of all laws and proceedings for the security of persons and property as is enjoyed by white citizens ...

And let's continue working backward further. Article III, Section 2 of the U.S. Constitution states that "The judicial power shall extend ... to cases ... between citizens of the same state claiming lands under grants of different states, and between a state, or the citizens thereof, and foreign states, citizens or subjects." (emphasis mine)

And now the final thing comes into play, and this might be the most important for you to consider. Under the earliest laws we have, it was assumed that American Indian tribes were sovereign governments ─ how else could the U.S. government sign treaties of land ownership with them? With Georgia and Alabama settlers encroaching on their land, and the state of Georgia prepared to impose its laws upon Indian land, the Cherokee Nation attempted to test this belief.

In Cherokee Nation v. Georgia (1831), Chief John Ross hired the Baltimore firm of William Wirt, who asked for an injunction "to restrain the State of Georgia, the Governor, Attorney General, Judges, justices of the peace, sheriffs, deputy sheriffs, constables and others, officers, agents, and servants of that State, from executing and enforcing the laws of Georgia, or any of these laws, or serving process, or doing anything toward the execution or enforcement of those laws within the Cherokee territory."

The court was very much split, with two conservative justices holding that the Cherokee Nation was not a nation at all. Two liberal justices held that the Cherokee Nation was just as much of a foreign nation as any other, and it deserved the injunction. Chief Justice John Marshall and two others took a compromise position that found the Indian tribes were "domestic dependent nations" with a relationship to the United States that "resembles that of a ward to his guardian."

This resolved nothing at all except to give the state of Georgia the legal fig-leaf it needed. It extended harsh anti-Cherokee laws over the Cherokee people. Meetings of the tribal council were forbidden, any non-Indian could enter Cherokee lands for any reason, and any Cherokee who acted as a judge in tribal courts was to be considered a criminal.

The Cherokees didn't give up. In Worcester v. Georgia, they forced the Marshall court to recognize that although the Cherokee nation was under the jurisdiction of the United States, the state of Georgia had no sway. Although Worcester v. Georgia is one of the foundations of Indian Law in the United States today, it was a Pyrrhic victory for the Cherokee. President Andrew Jackson refused to enforce it, and his refusal (and the subsequent passage of the Indian Removal Act) resulted in the Trail of Tears. It also opened the door for further persecution throughout the 19th century, as states east of the Mississippi successfully claimed to have jurisdiction over all tribes within their boundaries.

To summarize, Article III, section 2 of the U.S. Constitution permits "foreign states, citizens or subjects" to bring cases in federal court. The Cherokee cases enshrined that Indian tribes were "domestic dependent nations," which gave them the ability to bring cases in a way that black Americans (who would ultimately be bound by the Dred Scott decision) were not. If you're interested in the creation of American Indian law, I highly recommend Crow Dog's Case, which does a great job of walking you through the legal development of Indian law in the 19th century.