Very challenging! Not a common expression.
"Grace" in this context means "kindness". The sentence means that Chauncy is begging for food so that he and his mother can eat. Literally meaning that he/she is taking food off of other's kindness.
Another example - You are only successful, child, off (of) the hard work that your father did.
----- The child owes his/her success to their father's hard work in the past.
This is maybe older English that you would find in older novels. But I hope this helps!