Is it permissible to pay for the medical expenses of the husband's sister as fulfillment of a vow, or must the vow be fulfilled as originally vowed, by giving her the choice between 'Umrah, buying a stove, or a washing machine?

If you mean by the husband's approval merely his permission to give charity from his money which you do not own, then this is a vow concerning something not owned, and it is not valid, because of the Prophet's (peace be upon him) saying: "There is no vow upon the son of Adam concerning what he does not own."

However, if you intend to make the vow contingent on your husband giving you the money, then the vow becomes binding, and you must fulfill it if the condition is met and your husband gives you the money. If he does not give it to you, then nothing is incumbent upon you. It is disliked to make such a vow, even if it concerns something you own, because of the Prophet's (peace be upon him) saying: "Indeed, it does not avert anything; rather, it is used to extract [money] from the stingy person."

The fundamental principle is the obligation to fulfill a vow in the manner specified by its maker, and it is not permissible to deviate from it or change it without necessity.