Is it permissible for a son to fulfill a vow made by his father - who vowed to slaughter a sheep for Allah upon his son's employment - if the father is unable to fulfill the vow but the son is able to? And is the father sinful in this case?

Slaughtering a sheep out of obedience to Allah, and dedicating a vow of slaughter to a permissible matter, such as a son getting a job, necessitates fulfilling it. If the one who made the vow is unable to fulfill it, then he must offer expiation for breaking an oath. It is sufficient for him to authorize his son or someone else to fulfill the vow on his behalf. However, fulfilling it without his authorization is not sufficient, because it is an act of worship that requires intention or authorization.