What is the ruling on the vow made by the grandmother to perform Hajj on behalf of her grandson if God makes it easy, and is anything required of her?

Your grandmother's vow for her grandson to perform Hajj is not binding and she is not obligated to anything. A vow is "the commitment by a responsible, free-willed individual to God Almighty, through speech, to do something that is not originally obligatory by Islamic law." A vow cannot be binding upon another person, because that person is not under the authority of the one making the vow.