There are those who hold that a women place is in her home and therefor they shouldn’t drive. It was only in the 1920s that Bais Yakovs , the idea of Jewish female education, was born. Before that it was thought that anyone that taught their daughters Torah was a Heretic.
Similarly, in the last Mishna in Kiddushan 82A it says that a woman can’t teach yet all yeshivas have female teachers in the early grades.
Just as the Jewish people (Orthodox) couldn’t survive without female Jewish Education, so too they can’t survive without both parents working outside the house.
The question is , ” is driving a luxury or a necessity?” If you hold it’s a luxury then women shouldn’t drive but if you hold it’s a necessity then women must drive and just as a father is obligated to teach his son to swim so too must he teach his daughter to drive. Otherwise the husband will have to do all the shopping and cut into his learning time.
In an ideal society where food and clothing came down from heaven and as soon as you went to the bus stop or train station mass transit was there then perhaps woman shouldn’t drive.