All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
That one fears that he may fall down in the crowd when throwing pebbles or performing Tawaaf (circumambulation) or so cannot stand for a legal excuse that makes it permissible for him to deputize another to perform Hajj on his behalf. One can deputize another to perform Hajj on his behalf only if he is permanently incapable of doing it by himself.
Although it is valid, in the Sharee'ah, for one to deputize another to perform Hajj on his behalf, it is impermissible for the deputy to perform Hajj on behalf of more than one person by a single performance of Hajj. He can perform Hajj on behalf of each one with a separate Hajj.
Allaah Knows best.