Praise be to Allah.
The basic principle is that it is not permissible for a woman to travel without a mahram, whether the journey is long or short, because of the report narrated by al-Bukhaari (1729) and Muslim (2391) from Ibn ‘Abbaas (may Allah be pleased with him) who said: The Prophet (blessings and peace of Allah be upon him) said: “No woman should travel except with a mahram.”
See also the answer to question no. 110929
It is not permissible for a woman to travel with a driver who is not her mahram, if there is no one else with them, whether that is on a journey or otherwise, because this comes under the heading of khulwah (being alone with a member of the opposite sex), which is haraam. In the hadeeth it says: “No man should be alone with a woman without there being a mahram present.” Narrated by al-Bukhaari (5233); Muslim (1341).
See also the answer to question no. 10374
If what the questioner means is that there are other women with her in the car, in such a way that there is no forbidden khulwah (i.e., they are adults, of sound mind and so on), and that her mahram is also present during the journey, but he is in another car because there is no room for her in the car in which her brother is, or the group is spread out in such a manner that the men are in one car and the women are in another car during the journey, then there is nothing wrong with that, if the two cars are travelling together.
Shakh Ibn ‘Uthaymeen (may Allah have mercy on him) said:
Undoubtedly if a person travels with his mahram, such as his wife or a female relative, in the same car, that is better and is more on the safe side. But if there is a group of cars travelling together and making stops together, then there is nothing wrong with putting the women in one car and the men in the other car. But it is essential that the driver of the (women’s) car should not get separated from the car in which the male mahrams are, so that the mahram can keep an eye on the car in which his female mahram is.
Fataawa al-‘Uthaymeen, 21/216
But if what is meant is that the woman will travel on her own in the car with the driver, that is not permissible, because of what we have mentioned above about forbidden khulwah.
And Allah knows best.