Well, sort of. Ancestry.com has added U.S. Indian Census Schedules, 1885-1940.
These censuses cover only Native Americans who lived on reservations under federal supervision. The Constitution excludes these "Indians not taxed" from the federal decennial census, but a 2006 Prologue article discusses how this rule was bent in 1880 and 1890. And how poorly native names were recorded.
In many cases, only a single name, either the given or surname, is recorded. Frequently, enumerators recorded an English given name without any surname or used "Indian" as a surname or given name. In the 1880 census, for example, there are 924 enumerations in which the surname is "Indian" and 560 entries in which "Indian" is recorded as the given name with no surname. Occasionally, "papoose" or "squaw" is used for given names, or a number is used in place of the given name. One enumerator took the unusual step of making this note in the 1880 census: "Indians won't always give their names. When they do it is unsatisfactoryily [sic] given."