Women have really been driving in numbers since cars began to be produced. The earliest ones were seen as rich men's hobby-toys and required a certain amount of strength for steering, which made them less appealing to women, but very quickly they were a part of the market. But even as early as 1900, popular publications were referring to drivers as "men and women", testifying to the reality that wealthy women were driving alongside wealthy men. A long article in Our World Today (1904), "Automobiling for Women", noted that "the automobile has proved itself to be an emancipator for womankind", and went on to celebrate the accomplishments and habits of a number of female drivers who were "proving themselves the peers of men experts": Mrs. S. E. York, who drove with friends from St. Paul, MN to Portland, ME; Miss Dorothy Levitt, who received a perfect score in a run from Glasgow to London; Mme du Gast, who competed in the Paris-Bordeaux motor race; Mme Lockert, who drove from Paris to St. Petersburg; and many, many more.

In the early years of commercial automobiles, companies would sponsor women to drive across the United States as a kind of advertising stunt. The first was Alice Ramsey (with her sisters-in-law and a friend) in 1909, by this point already a veteran of the racing circuit; the car was given by the Maxwell-Briscoe Company, who also sent an advance man to promote her arrival in the towns she passed through. They also advertised electric cars to women for their relative quietness, and compared them to domestic contraptions like the stove or sewing machine to make more timid potential consumers more likely to try them. Suffrage organizations also ran cross-country promotional tours, and even etiquette maven Emily Post made her own personal trip from coast to coast.

Something else that should be remembered is that not long after the popularization of the automobile, World War I began. Despite the fact that only men could serve in combat roles, women actually contributed even on the front lines in a number of different ways - including as ambulance drivers. This gave women who already had driving skills an opportunity to use them for the war effort, and gave other women who might not have considered getting a car the experience to consider one in the future.

The really interesting thing is that despite women having been a part of the automobile scene from the first years of the twentieth century, the cultural stereotype that women weren't interested in cars and generally didn't drive them persisted so strongly. And while there were proponents of the idea that women are inherently bad drivers as soon as there were woman drivers, there were also doctors and insurance agents pointing out that women had many fewer accidents than men at the same time, and articles in popular publications working to defeat the prejudice.

Sources:

Christina Dando, Women and Cartography in the Progressive Era. (Routledge, 2017)

Katherine Parkin, Women at the Wheel: A Century of Buying, Driving, and Fixing Cars. (University of Pennsylvania Press, 2017)