Is it permissible for a son to perform Umrah on behalf of his deceased father, for whom an Umrah was assigned, and how should the intention be made if the name of the assignor is forgotten?

You may perform 'Umrah on behalf of the person who authorized your father, and the permission of that person or their heirs is not required; however, it is better to seek their permission. Also, it is not required to mention the name of the authorizing person when entering into the state of ihram. Rather, the intention that the 'Umrah is on behalf of the person who authorized your father is sufficient, for deeds are by intentions.