Is it permissible for a woman to ride in a car with a private driver to attend to her essential needs, accompanied by her children who are 8 and 4 years old, in the absence of her husband and without a mahram present?

It is forbidden for a woman to be in seclusion with a non-mahram (definition) man. This includes being alone with a driver in a car, even if it is traveling on public roads, because a car is akin to a private room. Seclusion is negated by the presence of a mahram for the woman, or by the presence of another woman with her, or by the presence of a discerning child (usually seven years old and above) whose presence would cause embarrassment. As for an undiscerning child, their presence is as if it were non-existent in preventing seclusion.