Can I accept and use the gifts and cash from my grandmother whose only source of income is bank interest? Please advice what Islamic law says about this.

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

It is Haram to receive any gift from your grandmother whose only income is bank interest. Also, scholars of Islam forbid dealing with such a person or accepting her gifts, since her wealth is Haram and she is not entitled to act with it freely. From Sharia point of view, she is not the real owner of such ill-gotten wealth.
Allah knows best.