Is it permissible for female teachers to travel daily a distance of one hundred kilometers without a mahram for the purpose of teaching, and is this considered a matter of necessity or need?

The distance mentioned in the question (100 km) is considered a travel distance by the consensus of jurists. It is not permissible for a woman to travel without a mahram (definition), whether the journey is long or short, due to the Prophet's ﷺ saying: "A woman must not travel except with a mahram." Accordingly, it is not permissible for the female teachers to move to that distance without their husbands or mahrams, and they must prove this to the employer to find a solution. If the responsible authority does not act, then this work must be abandoned by those who are not in dire necessity.