Is a single Udhiyah permissible on behalf of two people residing in two different cities and households, with their mother traveling between them, and is it sufficient for the mother to purchase the Udhiyah from her private funds on their behalf?

The Udhiyah (definition) (sacrificial offering) is a confirmed Sunnah (definition), or obligatory according to some jurists, for those who are capable. A single Udhiyah suffices for a man and his household, even if their number is large. The household includes the wife and children, and any relative who lives with the head of the household and is financially dependent on him or shares expenses with him. However, for someone who lives in a separate house and has independent expenses, a shared Udhiyah is not sufficient.