This is a matter of scholarly discussion. The majority of scholars from all four schools prefer that the sick হাজী be physically present for তাওয়াফ, even if they need to be carried, pushed in a wheelchair, or transported on a stretcher. তাওয়াফ is a physical act of worship closely tied to being physically present at the কাবা শরীফ, and the scholars prioritize the হাজী's presence.
Sheikh Ibn Baz stated that a sick হাজী should be carried for তাওয়াফ if possible — either on someone's shoulders, in a wheelchair, or on a bed/stretcher. He cited the report that নবী করীম performed তাওয়াফ on his riding animal during the Farewell হজ্জ. If the হাজী is unconscious, their family should carry them for তাওয়াফ, and the intention of the family member carrying them counts on their behalf. Only in truly extreme cases where the person absolutely cannot be transported does the question of proxy তাওয়াফ arise.
Sheikh Ibn Uthaymeen held that proxy তাওয়াফ (where the proxy goes alone without the sick person) is not established from the Sunnah and is a weak position. He strongly মুস্তাহাব making every effort to transport the sick person, noting that modern wheelchair services at the Haram make this much easier than in previous eras. If the sick person is in a hospital and cannot be moved at all, they should wait until they recover to perform their তাওয়াফ, as there is no time limit for তাওয়াফ al-Ifadah. In extreme cases where recovery is not expected, some scholars from the Hanbali school permitted proxy তাওয়াফ by analogy with proxy হজ্জ.