Is it permissible for employees in the Public Relations department to take a discount percentage from the supplier of Eid cakes in exchange for their extra effort and with the approval of their superiors?

It is not permissible for you to take the discount percentage for yourselves unless the government administration grants permission, because you are agents of the company, and the discount is its right. You can request compensation from the company for your extra work. It is not lawful for an agent to take a commission without the knowledge and consent of the principal. Even if your superiors know about it, it does not make it permissible for you unless they are authorized by the administration to grant permission.