All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
It is permissible for her to drive the car if she can drive it properly without incurring harm on herself or others. She is not obliged to obtain a new driving license if she is now entitled to have one. It is sufficient for her to repent for offering the bribe and her genuine remorse for committing that sin is the sign of the sincerity of her repentance and return to Allaah. She should regret her sin and resolve never to commit it again in the future.
For further benefit, please refer to Fatwa 18289.
Allaah Knows best.