Is it permissible for a woman performing Mut'ah Hajj and Umrah on behalf of her father to wear socks or slippers during Tawaf and Sa'i?

It is permissible to perform 'Umrah on behalf of your deceased father, and its reward will reach him, Allah willing. However, if he is alive, it is a condition that he be permanently incapacitated in a way that recovery is not expected, that he grants you permission to perform it on his behalf, and that you have already performed 'Umrah for yourself first. It is permissible for a woman to cover her feet with socks or khuff (leather slippers) while in a state of ihram.