Is the family obligated to pay Zakat al-Fitr from what is surplus to their needs on the eve of Eid and on the day of Eid, if that amount is not sufficient to pay the Zakat for all its members, or is the Zakat waived for them in this case?

Zakat al-Fitr is obligatory upon every Muslim who possesses an excess beyond their basic needs for the day and night of Eid, [amounting to] one sa' of their staple food. Zakat al-Fitr is given from what remains after covering the basic needs of one's dependents, as stated in the Hadith: "Begin with yourself and give charity to it; if anything remains, then for your family."

A Muslim is required to pay Zakat al-Fitr for themselves and for those they support, such as a spouse, children, and relatives. If what one possesses is insufficient to pay for everyone, one should start with oneself, then their spouse, then their mother, then their father, then their children, then the closest relatives in terms of inheritance. If two or more individuals are equally close and only one sa' remains, lots should be drawn between them.