What is the ruling on one who vows to slaughter a chicken (with the intention of slaughtering a sheep) if his debt is paid off, and is it permissible for him and his family to eat from it?

This oath is binding on righteousness as long as what you conditioned it upon has not occurred. So, when the money is found, you are obligated to slaughter the sheep you intended, because deeds are by intentions, and the intention of the swearer particularizes his oath. If you did not specify a particular destination for the slaughtered animal, then its recipients are the poor and needy, and you should not eat from it unless it is customary in your country for eating from it to be permissible in such cases.