Let's start by clarifying the historical geography: What is today the state of Oklahoma is the result of three leftover pieces of territory. The eastern part of the state was reserved for the "Civilized" (aka slave-owning) tribes pushed out of the American Southeast. The western half was later divvied up to other tribes (eg: the Osages) as they got pushed out of their territories. Generally, these later tribes weren't force-marched into their new territories by the army though, so the tribal element in the western reservations wasn't nearly as strong as in the east. The "panhandle" was part of Texas, but was too far North to fit in the slave state/free state compromise that allowed Texas to join the Union, and too far south to fit in a similar compromise that brought in Kansas.

Indian Territory (the eastern half of Oklahoma) was originally supposed to be land set aside for the "Five Civilized Tribes'" use. Each tribe had its own specific territory which it was supposed to govern.

The problem was white folk didn't tend to respect tribal governments, so effectively the Indian areas became lawless areas; a haven for outlaws and squatters. There was even a quasi-organized movement of White land thieves, called "boomers"

Meanwhile, the territory was rather attractive to African Americans as a place where they could go to live their lives away from the micro and macroagressions of racist white-run governments. It didn't hurt that the tribes themselves brought over their slaves, so there were in fact already a lot of "Black Indians" living there.

As a proud native Okie myself, I like to think of us as the land nobody wanted, filled with the people nobody wanted.

Eventually the US government stepped in, made it a proper territory, and forced all the tribes to divvy up their reservations to their individual members. (As a result of this, Oklahoma technically has no "reservations", unlike a lot of states with significant Native American presence)^{(see "update" below)}. Excess land at this point (aka: Unassigned Lands) was opened up to white settlement. Also, many of the individual tribe members didn't live very close to their assigned plots, weren't very well-versed in USA property law, and thus were easy prey for white boomers and speculators. So by the beginning of the 20'th century native tribe members actually didn't own much land in "Indian Territory" at all.

Another thing to realize here is that traditional tribal culture viewed tribal and clan affiliation as voluntary, rather than hereditary. In other words, they didn't have USA/European culture's obsession with "race" or "blood". People would move between tribes at will, and tribes and clans had little compunction about adopting white (or black) people who were serious about joining the tribe. Indeed many of the tribal leaders on the Trail of Tears were by our reckoning mostly White. For example, Cherokee Chief John Ross was 1/8th Cherokee and 7/8ths Scots, and his first language was English. So when looked at with our modern White/Red/Black blinders on, who went where becomes very confusing.

So yes, there were certainly many "white" (and "black") Americans living in Indian Territory.

Update

A decade after I wrote this, an important part of it became obsolete. Specifically, while the Federal and State governments had been acting as if all the reservations in the state had been disestablished, in 2020 the Supreme Court ruled that Congress had never actually done so, and refused to do the dirty work for them in court. This means legally, there are now and have always been several "reservations" in Oklahoma. This so far has been shown to include the Quapaw and all five "Civilized Tribes", but probably includes nearly every American Indian tribe in the state.

Further Reading

For an actual historical work covering the lawlessness of Indian Territory, I found Black, Red, and Deadly: Black and Indian Gunfighters of the Indian Territory, 1870-1907. It's mostly a compilation of period newspaper excerpts, but boy are some of them interesting.

For a heavily fictionalized (in fact somewhat fantastical) treatment of the same characters, time, and place, see the movie The Harder They Fall. Sadly, it almost completely ignored the Native element of the territory, but otherwise a great flick.

A wonderful book on the culture of the "civilized tribes" in the early 1800's and how that culture interacted with the racial attitudes in the United States culture is a work of alternative history titled 1812:The Rivers of War. The sequel isn't as strong on this particular subject, but is one of my personal contenders for best work of fiction ever. The author was working on a 3rd installment of the series when he passed. I don't know if it will ever see the light of day.