The proverbial phrase “to set foot under someone’s table” meant to be financially dependent on that person. It evokes the image of an elderly parent forced by financial necessity to move in with one of the grown children, and take his meals at the family table. Gremio’s doubts that Lucentio’s father — remember that Tranio is pretending to be Lucentio — would be willing to give all his wealth to his son, and in the process put himself in that position.