[Is Hajj Valid on Behalf of an Elderly Sick Woman Who Tires from Walking?]

Answer: If the matter is as described, it is permissible to appoint someone to perform Hajj on her behalf. One whose arm and leg are paralyzed—especially alongside advanced age and weakness—cannot manage ṭawāf and saʿy amidst the severe crowds. Indeed, it is not permissible for such a person to endanger herself in the crowd: “Allah does not charge a soul except [with that within] its capacity.” [al-Baqarah]
If it is said: There are wheelchairs/carts by which ṭawāf and saʿy can be done for a fee.
It is said to him: That is not required—even if such chairs are available—because the excuse remains: paralysis of an arm and a leg together with the weakness of old age.
Moreover, what appears to me—and Allah knows best—is that a very elderly woman, and likewise a very elderly man whose strength has been sapped by age, may validly have Hajj performed on their behalf during their lifetime, on condition that they acquire a strong presumption and conviction that they cannot perform Hajj. When that presumption of incapacity is attained, it is permissible for them to commission someone to perform Hajj on their behalf.
Source : Min Thimār al-ʿIlm wa al-Ḥikmah vol.1