Is it permissible for me to take from my grandfather's money, given his stinginess towards my father and us, without his knowledge, and is my grandfather considered sinful for not fulfilling his duty towards his children?

It is not permissible to take anything from the grandfather's money without his permission. He is not obligated to provide for a grandchild who is able to earn a living or who is supported by his father. However, if the father is poor and cannot find enough to suffice him, then the wealthy grandfather is obligated to support him, according to the Hanbalis. The grandfather sins if he withholds support from his needy son, for the Prophet, peace and blessings be upon him, said: "It is sufficient sin for a person to withhold food from one whose sustenance he possesses."