Can Women Do Tawaf without a Mahram?

It is permissible for a woman, if she is in Makkah, to go walking from the hotel to the Haram without a Mahram , and to do Tawaf likewise, because it is only necessary to have a Mahram for travelling ; inside the city it is not essential to have a Mahram. Rather what is stipulated is that a woman should feel safe in going out and not to wear adornment or make-up when she  goes out. 

The same may also be said concerning her going to stone the Jamarat. She may go there without a Mahram; she may go on her own or with a group of women. 

If a woman is in one city and wants to go somewhere in a taxi, it is stipulated that she should not be alone with the driver ; if there is a righteous woman with her, then she is not alone with him. 

Shaykh Ibn `Uthaymin (may Allah have mercy on him) said: 

“With regard to Khulwah (being alone with a member of the opposite sex) inside the city, it is not permissible for a woman to be alone with the driver in a car, even for a short distance, because the Prophet (blessings and peace of Allah be upon him) said: “No man should be alone with a woman without a Mahram being present.” But if there is another woman with her and the driver is trustworthy, then there is no Khulwah and there is nothing wrong with her and the other woman travelling in the car, so long as their trip is not regarded as “travel.” In this case, we may say that there is no Khulwah because of the other woman who is present. But we cannot say that the woman who accompanies her is regarded as a Mahram; rather we say that what is forbidden inside the city is a man being alone with a woman, unlike travel, in which case it is forbidden for a woman to travel alone without a Mahram. The difference between these two matters is clear.” (Majmu` Fatawa Ibn `Uthaymin, 21/191) 

And Allah knows best.