When Dartmouth College named Susan Taffe Reed as the new director of its Native American Program in September, it said she would use her role as president of the Eastern Delaware Nations to help Indian students adjust to life at the Ivy League school.

But since Ms. Reed’s hiring, tribal officials and Dartmouth alumni have accused her of misrepresenting herself as an American Indian and the school of failing to adequately vet her background as a leader of a nonprofit group that acknowledges it isn’t an actual tribe.

Ms....