In E’Y the women Litvish probably dont drive because its too expensive. In Lakewood the women most certainly do drive.
In Boro Park I think quite a few chassidic women do drive.
There is a distinction between living in Williamsburg where everything is quite close to each other, there is traffic congestion. And you can walk or talk Public transport fairly easily and KIryat Joel where things are further apart and a car is much more nessasary and it is a hinderance for people especially with many kids not to drive