First off, 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 farming) tribes pushed out of the American Southeast. The western half was later divied up to other tribes (eg: the Osages) as they got pushed out of their territories. Generally they weren't force-marched into their 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.
As for the Indian Territory (eastern half of Oklahoma) itself, it was originally supposed to be land set aside for the tribe's 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.
Eventually the USA government stepped in, made it a proper territory, and forced all the tribes to divvy up their reservations to their indivdual members. (As a result of this, Oklahoma technically has no "reservations", unlike a lot of states with significant Native American presence). 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 speculators. So by the beginning of the 20'th century 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 the 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 Cheif 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 certianly many "white" (and "black") Americans living in Indian Territory.