Is it obligatory to make up a vow to fast for nine consecutive months, knowing that the vow was made to cease committing a sin, and that the Prophet, peace and blessings be upon him, forbade continuous year-long fasting?

Making a vow is disliked, but whoever vows an act of worship, such as committing to fasting, must fulfill it. This is based on the Almighty's saying: "And let them fulfill their vows," and the Prophet's (peace be upon him) saying: "Whoever vows to obey Allah, let him obey Him." Scholars have affirmed the validity of vowing to fast for a year. The prophetic prohibition of voluntary fasting does not apply to fulfilling vows. It is obligatory to fulfill the vow according to what you intended regarding the continuity or separation of the fasting. If you intended continuity, then it is binding upon you; otherwise, it is not. The continuity of the eight months is not binding unless you specified them as particular, successive months in the vow. In that case, their continuity is obligatory, and it is not permissible to delay them while being able. Deliberate delay incurs sin and an expiation for breaking an oath (kaffarat yamin). If the months are not specified, then it is recommended to hasten fulfillment, but it is not obligatory. It is not permissible to deviate from the fasting unless you become unable to perform it with an inability that is not expected to cease; in such a case, you owe an expiation for breaking an oath.