Is the wife sinful for not fulfilling her vow to bring the husband's mother to perform the Hajj pilgrimage with them, or is the husband sinful, and what is obligatory upon them both regarding this, knowing that the mother has previously performed the pilgrimage and her health does not permit travel and Hajj?

The questioner should be advised of two matters: First: The phrase (my wife vowed to send...) suggests an intention of vowing to other than Allah, which is polytheism (shirk). The correct phrasing is: (my wife vowed to Allah that she would send...). Second: The vow is binding on the wife unless the husband approves it, based on the Prophet's (peace and blessings be upon him) saying: "A man is not bound by a vow concerning what he does not possess."

And the answer to the question: - If the mother is able to perform Hajj completely, or with deputization where permissible, then fulfilling the vow is obligatory if the financial means are available. It remains a debt until the means become available. - However, if the mother is genuinely unable, with no hope of recovery, then the expiation for the vow is the expiation for an oath (kaffarat yamin), which is feeding ten poor people, or clothing them, or freeing a slave. If one cannot find these, then fasting for three days, based on the Prophet's (peace and blessings be upon him) saying: "Whoever vows a vow he cannot fulfill, its expiation is the expiation of an oath. And whoever vows a vow he can fulfill, let him fulfill it." - The ability to perform tawaf and sa'i on foot is not a condition, as there is an alternative of riding. Nor is direct stoning (rami) in crowds a condition, as it is permissible to delay it or have someone else do it on one's behalf.