Is it permissible to substitute a vow to go out for three days in the cause of Allah with fasting for three days?

A vow does not change destiny. If Allah has ordained success for the sister, she will succeed whether she makes a vow or not. As for the ruling on her vow to go out, if she intended it for learning, dedicating herself to worship, and da'wah (inviting to Islam), and no forbidden act results from it, then she must fulfill it and is not permitted to renege on it. If an impediment arises that is expected to be removed, the vow remains binding on her until she is able to fulfill it. If the impediment is not expected to be removed, then she must offer an expiation for an oath (kaffarat yamin) due to her inability.

Ibn Qudamah states that one who is unable to fulfill their vow due to a temporary excuse should wait for its removal. If the inability persists and is no longer expected to be removed, then they must offer an expiation for an oath. Whoever vows an act of obedience that they cannot perform or become unable to perform, they must offer an expiation for an oath. Imam Malik confirms in Al-Muwatta' that a woman's vow is valid even without her husband's permission, as long as it does not infringe upon his rights. If it does infringe upon his rights, it remains binding on her until she can fulfill it without harming him.