They could and did. There's a long history of indigenous slavery, indentured servitude and related labor systems and following the American Revolution some Native nations adopted the US system of slavery. For obvious reason, this was more common for the nations in the South than those in the North. Even the Seminoles, who famously offered refuge to escaped slaves, had slaves of their own - though they enjoyed considerably more rights than those owned by white Americans.

For the Cherokee, from 1809 to 1835, the slave population grew from 583 to about 1600 - about a tenth of the Cherokee population itself at the time of Removal. Though, of the 16,000 or so Cherokee, only 209 where slave owners in 1835. During removal, both the Cherokee and their slaves were forced across the Mississippi. The Cherokee slave owners in Oklahoma led a vocal minority of the Cherokee into an alliance with the Confederacy, with the preservation of slavery being one of the key reasons for the alliance (other factors included the promise of a representation in the Confederate Congress and an "enemy-of-my-enemy" stance regarding the federal government). After the Civil War, all slaves were freed, including those owned by Native Americans. The Cherokee slaves became the Cherokee Freedmen who remained part of the Cherokee Nation until relative recently -- but that's an ongoing issue that should be hitting the Supreme Court this year. If you want to know more about that, maybe someone over at /r/IndianCountry can catch you up on current events.