While very few of these records have been digitized and made available online, some records relating to enrollment or censuses of Native Americans at the National Archives include the Dawes Rolls, the Eastern Cherokee rolls, and the US Indian Census rolls.
Most of the microfilms of these records are now available on Fold3.com, accessible for free on computers at regional National Archives branches, or for a small fee from home. In some instances it may be more affordable to do research on Fold3 than to order copies from the National Archives.
When searching the microfilms, you can search either by name, or look at the roll itself. This includes the Dawes Rolls, the Eastern Cherokee rolls including Guion Miller and earlier rolls, and the US Indian Census rolls. You can see the applications to the Eastern Cherokee and make copies from them. You can access the doubtful and rejected applications, as well as the accepted ones.