Is it permissible to break one's vow fast – Mondays and Thursdays for life – if one falls ill, and is it obligatory to make up for the days that were broken due to fatigue, illness, or menstruation?

A vow of obedience must be fulfilled, for the Prophet, peace and blessings be upon him, said: "Whoever vows to obey Allah, let him obey Him; and whoever vows to disobey Allah, let him not disobey Him." Narrated by Al-Bukhari. There is no sin in not fasting during illness or menstruation, but these days must be made up.