Is there a specific way to initiate a vow (nadhr), is it permissible to make a vow contingent on the marriage of a specific person, and must this vow be fulfilled if the wish is not granted?

A contingent vow is disliked, and it does not avert anything. However, it must be fulfilled if the condition upon which it is contingent is met, and the one who made the vow is not liable for anything if the condition is not met. The proof for the obligation to fulfill it is the saying of the Prophet, peace and blessings be upon him: "Whoever vows to obey Allah, let him obey Him," and the general meaning of the Almighty's saying: "O you who have believed, fulfill [all] contracts," and Allah Almighty's praise for those "who fulfill vows." A vow is the commitment of a legally responsible person to an act of worship to Allah that is not originally obligatory. It requires a صيغة (formulation) that expresses commitment, and prayer is not prescribed for the sake of a vow.