The earliest instance "of put on a clinic" that I've found in various database searches is from "Missouri-Kansas-Oklahoma Ass'n's Annual Meeting,"in The Hotel Monthly (December 1915):
J. P. McGuire, of Armour & Co., put on a clinic, using a yearling wether mutton carcass, which an expert butcher dissected into market cuts, the lecturer explaining as the cutting was done. This clinic will be the subject of a special article in THE HOTEL MONTHLY.
Also, from "Report of Oral Hygiene Committee O[ntario] D[ental] S[ociety]," in Oral Health (May 1919):
Mr. Putman, of the Agricultural Department of the Provincial Government, asked for the co-operation of the Committee to make an investigation of the school children of Streetsville, and put on a clinic for two days. The Committee complied with the request and Dr. Bothwell kindly took charge of this work.
And from "Symposium on Health, Labor and Education—Infant Welfare Clinics," in Proceedings of the first state conference of child welfare boards with the Board of Control. State Capitol, May 9 and 10, 1919, St. Paul, Minnesota (1919):
Mrs. Johnson: We attempted to put on a clinic without outside help, the doctors in town giving their services, and 406 children were brought in. Then the county welfare board took the list of defective children and tried to follow it up.
To sum up, an instance of "put on a clinic" from 1915 involves demonstrating professional mutton butchering at a regional hotel association meeting in St. Louis, Missouri. An instance of the phrase from 1919 refers to providing dentistry services to the children of a town in Ontario. And a second instance from 1919 involves an ad hoc effort to test children in St Paul, Minnesota, for tuberculosis and other diseases.
A fourth instance, from slightly later in 1919, appears to be the one cited in DjinTonic's answer. From "The Pueblo County Dental Society," in Bulletin of the Colorado State Dental Association (December 1919):
Regular meetings [of the Pueblo County Dental Society] are held the third Thursday of each month, usually at one of the offices, or preceded by a dinner at one of the hotels. A committee has been appointed to put on a clinic about March first.