What is the ruling on a woman leaving her home to buy groceries and the like, with the approval of her mahram who is unable to accompany her due to illness, sleep, or work?

There is no blame on a woman for going out to buy her necessities if she adheres to the Sharia guidelines, such as modesty and refraining from using perfume. The presence of a mahram is not a condition for her to do so. However, for a married woman, her husband's permission is required for her to go out, and she should avoid marketplaces if there is no need. Furthermore, her riding alone in a car with a non-mahram man is considered an unlawful khalwa (seclusion).