Is it necessary to sit while supplicating with the prayer of al-Ta'ar, or can one supplicate while lying down on the bed?

You are not required to sit down to supplicate (make du'a) when you awaken during the night. It is permissible for you to supplicate while lying on your bed or in any other state you happen to be in. It has been authentically reported from the Prophet, peace and blessings be upon him, that the Companions would remember Allah and magnify Him while they were in their beds. Similarly, he, peace and blessings be upon him, would recite (Qur'an) and supplicate before sleeping while reclining.