Most certainly. Especially for those American Indian units which sides with the Confederacy, they were left almost entirely to their own devices when it came to equipment, leading to little in the way of uniformity to say the least, and most warriors simply taking whatever they felt like to war with them, both for weapons and equipment. Bowie Knives seem to have been the most popular of 'close-combat weapon', not dissimilar to the blade preferences of many non-Natives, but certainly other weaponry such as tomahawks were not uncommon, and at least some accounts of interactions with indigenous soldiery mention war clubs too.

But that said, it shouldn't been understood as "Regiment [X] used tomahawks". Some soldiers did, and some didn't. It was more a matter of personal preference than anything else, and it should more be said that there was no barrier to their use if they wanted. And of course, the flipside is that the imagery of 'tomahawk wielding "Injuns"' was one thrown about to denigrate the behavior, rightly or wrongly, of Native American units during the war in a way that bad behavior by white units wasn't criticized.

Native American Mounted Riflemen 1861-65 by Mark Lardas

The American Civil War in the Indian Territory by John D. Spencer

The American Indian as Participant in the Civil War by Annie Heloise Abel