Is it permissible to buy the niqab with the mother's money without her knowledge, given that she forbids it, and is one held accountable for not wearing it when there is a impediment, despite believing in its obligation?

Obedience to parents is obligatory in what is good and proper, but their obedience is not obligatory in abandoning the niqab if you believe it is obligatory. If you do not possess money, it is permissible for you to buy the niqab with their money without their knowledge, because you have a right to maintenance according to what is customary. If you are unable to cover your face in front of non-mahram men due to their prohibition, then there is no blame upon you, but do not go out or appear before non-mahram men except out of necessity. Your parents have a right over you, no matter their condition, so be patient and be good to them.