Is it obligatory to fulfill a vow of fasting, or is it better to give charity, and is a vow contracted by intention or must it be uttered?

Whoever vows obedience to Allah must fulfill it, for the Prophet, peace and blessings be upon him, said: "Whoever vows to obey Allah, let him obey Him." The vow must be fulfilled as it was made, and changing it is not permissible. Therefore, your aunt must fast what she vowed if your grandmother recovers, and giving money as charity instead of fasting is not sufficient. A vow is not binding merely by intention, and undertaking a vow is disliked because it is forbidden.