You're correct that the LPGA was formed in 1950. But the game of golf certainly existed long before then. Using the term loosely, professional
could apply to anyone who makes a living doing something. In Jordan's case, if she played golf in tournaments, taught the game to others, or derived her primary income from golf, she could still be considered a professional without belonging to any association.
Today's world is much more likely to have professional organizations tied to most fields. It's unlikely you'll meet anyone who claims to be a "professional anything" who doesn't have some certification or professional society backing up that claim. Such wasn't the case in the 1920s, in Gatsby and Jordan Baker's world.