In 1983, Glenn received the first national award of the American Speech and Hearing Association for her meritorious service to those with communicative disorders. In 1987, the National Association for Hearing and Speech Action awarded the first annual Annie Glenn Award for achieving distinction despite a communication disorder. Glenn presented the award to James Earl Jones as its first recipient. She was inducted into the National Stuttering Association Hall of Fame in 2004. In 2015, The Ohio State University renamed 17th Avenue (on its campus) to Annie and John Glenn Avenue.
Glenn was played by Mary Jo Deschanel in the 1983 film The Right Stuff. The film highlighted her stutter, particularly in a scene involving U.S. Vice President Lyndon B. Johnson. In a 2015 interview, she and John Glenn indicated that, although they liked the Tom Wolfe book, they did not care for the movie adaptation of The Right Stuff.