Is it permissible to pay a sum of private money to someone to perform Hajj on behalf of the deceased wife's father who did not perform Hajj or Umrah, or must it be from the children's money?

It is permissible to perform Hajj and Umrah on behalf of a deceased Muslim by anyone, whether a relative or a distant person, provided that the proxy has already performed Hajj for himself. The proxy may assume any type of ritual (Tamattu', Qiran, or Ifrad) unless a specific ritual is stipulated. It is not a condition that the funds come from theceased or their heirs.