I just took the quiz on The Great Gatsby on this site. How can Jordan Baker be described as a professional golfer? To my knowledge, the LPGA did not form until the mid-1950s. Shouldn't she be referred to as an amateur golfer instead?

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.