Is it permissible to spend money vowed for the sake of Allah on incentive competitions for reading books of Islamic law?

A vow of obedience must be fulfilled. If, when you made the vow, you intended a specific path with your words: "for the sake of Allah," such as jihad or giving it to the poor, then you must spend the money on what you intended. If you did not intend a specific path, then it is permissible to spend the money on any act of obedience to Allah the Almighty, such as spending on da'wah (calling to Islam), on seeking knowledge, and on its students. There is no harm in fulfilling the vow in the form of monetary prizes for winners in the pursuit of knowledge.