The trouble with answering this question is that the "flappers" were not any kind of organized or unified movement. The present-day conception of the flapper is largely based on post-period nostalgia - in reality, the "flapper" label was not unlike the use of "Millennial" today: originally applied as a label of age/cohort, and eventually developing into a stereotype that is rarely fully embodied in any one person.

You can see this previous answer of mine for a bit on the most likely etymology of "flapper", in my opinion - if you look at other answers there, you can see there are other opinions. Regardless of the ultimate origin, by the first two decades of the twentieth century (1900-1919) it was in common use to mean "a teenage girl." To quote from a blog post of mine:

Both women's and men's lives were divided up into strict partitions: infancy, childhood, girlhood and boyhood, womanhood and manhood. While someone was in girlhood, she was expected to behave as a girl and to be treated as a girl, not an adult. Once she was a woman, she would change her whole manner of dress (usually, longer skirts and done-up hair) and deportment. Flappers upset this order - they didn't fit into it. They were meant to act like girls even though they looked like women. The Theatre Magazine's article on fashion for flappers focuses on how parents can dress their daughters to avoid putting them into women's dress too soon, signalling that they were adults ready to be married, while also avoiding making them look ridiculous in too-girlish clothes that were physically wrong for them to wear.

During and after World War I, the term became more nebulous - sometimes still referring to teenage girls, sometimes referring to leisured and affluent young women who had no expectation of working, and sometimes referring to the stereotype we have today of older young women of marriageable age. Because this was just a term to apply to young women rather than a movement along the lines of the 1960s feminist movement or earlier activists for "woman suffrage", it's not like there was an equivalent to Gloria Steinem or Elizabeth Cady Stanton - there were no literal leaders. Are you interested in reading about famous figures that may have influenced young women of the period, or that would have strongly represented the flapper "type" at the time? That's not quite the same thing, but it's similar.

(I will say that Irene Castle initiated one of the biggest fashion trends associated with the flapper stereotype: bobbed hair. While the development of the slender, girlish fashionable figure of the 1920s came gradually in high fashion through the choices of dozens of designers, the bob appears to have come about as a direct result of Irene Castle's cutting her hair in 1914 or 1915 while she was ill. Though she hid the cut at first, at the urging of friends (reportedly) she wore it openly, and the "Castle bob" and ways of achieving it appeared in magazines from mid-1915. Here is a publicity shot of her from that year, sporting her bob. Irene was, I should have said, a very famous dancer of the period: she and her husband, Vernon, appeared in a number of films and stage shows during the 1910s and ran a dancing school.)