Is it permissible for a sister to drive a car that she bought with money she earned from her work as a women's hairdresser?

If your sister's work is permissible, such as beautifying women under the prescribed conditions, then her money is lawful, and there is no harm in you using her car. However, if her work is forbidden, then the earnings from it are unlawful. Nevertheless, it is not forbidden to borrow the car purchased with this money, because the unlawfulness is tied to the responsibility of the seller or donor, and not to the purchased item itself.