All perfect praise be to Allah. Blessings and peace be upon Prophet Mohammad and upon his family and companions.
There are two situations in which it is permissible to authorize someone to do Hajj ( Major pilgrimage ) on behalf of another:
First: If Hajj was due on a Muslim, but he died before performing it, his heirs should pay someone from his estate to perform Hajj on his behalf, and it is also permissible for a Muslim, an heir of the deceased, or not, to perform it on his behalf on voluntary basis.
Second: If a Muslim was inflicted with an incurable disease, or was too old to travel, and was financially capable to cover the expenses of Hajj, then he should authorize .someone to perform it on his behalf even with paying him a fair. And Allah Knows Best.