All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
It is permissible for a woman to vow to fast without her husband's permission, and the vow is valid and binding. In principle, both men and women are equal in terms of the rulings of the Sharee'ah, unless there is evidence specifying her with a certain ruling, such as disallowing a woman to observe a voluntary fast when her husband is present unless he permits her. In Al-Muhalla, Ibn Hazm said:
"A man, a virgin woman with or without a father, a woman with or without a husband, a slave, and a free person are all equal regarding the fulfillment of a vow. Allaah Almighty commanded the fulfillment of vows and so did the Prophet, sallallaahu ‘alayhi wa sallam, without specifying or excluding anyone. Allaah says (what means): {And never is your Lord forgetful.} [Quran 19: 64]"
This is the Sharee'ah ruling regarding making a vow. As for the fulfillment of the vow, there are some details in this regard. In Al-Muwatta‘, Imaam Maalik said:
"According to our view, the woman's vow without her husband's permission is permissible, and she is required to fulfill it. If she vowed to do a physical act of worship and the fulfillment does not harm her husband, then she has to do it immediately. If it causes harm to her husband, then she is not absolved of liability unless she fulfills the vow."
Commenting on Maalik's statement, Al-Qurtubi said in Al-Muntaqa:
"A woman's vow to do a physical act of worship, such as praying, fasting, and performing Hajj, is of two types: firstly, a vow which causes harm to her husband, such as frequent fasting and Hajj (pilgrimage). Secondly, a vow that does not cause harm to her husband, such as performing two Rak‘ahs (units of prayer) or fasting a day. If the fulfillment of the vow causes harm to her husband, then he has the right to disallow her because he has the right to enjoy her at anytime, so she is not allowed to do anything that prevents him from this right. However, she remains liable to the vow until she finds an opportunity to fulfill it. If the vow does not cause harm to the husband, then she can immediately fulfill it and her husband should not prevent her."
At the end of the answer, we alert you that making a vow is disliked in the sharia in principle.
Allaah Knows best.