Is my mother required to fulfill her vow to fast for four consecutive months, which she made when she was fourteen or fifteen years old upon seeing her sister in childbirth, given that she was ignorant of the Sharia rulings at that time?

Most likely, the mother had reached the age of responsibility (taklif) during that period, and she is obligated to fulfill her vow as long as she is able, due to Allah's saying: {and let them fulfill their vows} (22:29), and the Prophet's (peace be upon him) saying: "Whoever vows to obey Allah, let him obey Him." If she is completely unable to fulfill the vow, then she must offer an expiation for an oath. However, if she was ignorant of the meaning of the word she uttered, then she is excused by her ignorance and owes nothing.