Leonard Wright Edwards attended Southern Normal University in Huntingdon, Tennessee, and graduated from Vanderbilt Medical School with his M.D. in 1912. Following graduation, Edwards was appointed part-time demonstrator in surgical anatomy and assistant in the Department of Surgery at the Vanderbilt Medical School. Soon afterward, he joined renowned surgeon and Vanderbilt clinical professor Dr. Duncan Eve in private surgical practice.

Edwards served as a first lieutenant in the Medical Corps during World War I and taught at the post-graduate school organized at his post in Alabama before reassociating with Eve in 1938, forming the Edwards-Eve Clinic. A pioneer in vagotomy-antrectomy surgery, Edwards is responsible for the establishment of an approved residency training program in surgery at St. Thomas Hospital in Nashville. He was also active in many local and state medical associations.