What is the ruling on one who vowed to pray Fajr in the the mosque but was unable to do so due to illness or a long journey?

A vow is not binding unless it is in the form of a commitment, and the mentioned wording is not a form of vow. If there were a form of vow, and the person made an exception for the state of illness or an emergency, then nothing would be binding upon him for missing prayer in the mosque due to illness or the aforementioned reason.