After the death of my father, who vowed to slaughter a sheep but did not fulfill his vow, is it better to slaughter on his behalf, or to give the value of the sheep as ongoing charity for him?

It appears that the wording mentioned in the question is not a form of vow (Nadhar), but rather a promise. Consequently, nothing is obligatory due to the absence of an explicit vow formula. However, fulfilling the promise is recommended. There is no objection to fulfilling it on behalf of your father as a precaution, on the basis of recommendation, not obligation.