Is it permissible for a person to perform Umrah on behalf of a deceased person without taking permission from their heirs, or is it a condition that he seeks their permission?

It is permissible to perform 'Umrah on behalf of the deceased without the permission of their heirs, because performing 'Umrah on their behalf is like supplicating for them and giving charity, which does not require the consent of the heirs. It is a condition, however, that you must have first performed 'Umrah for yourself.