Can You Stone the Jamarat on behalf of Someone Else?

If the person who delegated you to stone the Jamarat on his behalf is unable to do it because of the hardship and crowding, it is permissible for you to stone them on his behalf . You mentioned that he is elderly, so it is most likely that he is unable to do it himself . 

It says in Fatawa al-Lajnah al-Daimah, 11/286. 

“If a person is unable to stone the Jamarat, then he may appoint someone to do it on his behalf . Jamarat al-`Aqabah and the others are all the same in this regard. He should appoint someone who is trustworthy and is also doing Hajj that year… 

Based on this, your stoning the Jamarat on behalf of that elderly person is acceptable and you do not have to do anything. 

Each time you stone the Jamarat, you should stone it first for yourself and then on behalf of the other person, and you should do this at each of the Jamarat. You do not have to stone all three first for yourself and then go back and stone them on his behalf, rather you can stone each one for yourself and then for him, in one standing . But you should start by stoning it for yourself. 

And Allah knows best.