This is a matter of scholarly discussion. The majority of scholars from all four schools prefer that the sick jamaah haji be physically present for Tawaf, even if they need to be carried, pushed in a wheelchair, or transported on a stretcher. Tawaf is a physical act of worship closely tied to being physically present at the Kabah, and the scholars prioritize the jamaah haji's presence.
Sheikh Ibn Baz stated that a sick jamaah haji should be carried for Tawaf if possible — either on someone's shoulders, in a wheelchair, or on a bed/stretcher. He cited the report that Nabi performed Tawaf on his riding animal during the Farewell Haji. If the jamaah haji is unconscious, their family should carry them for Tawaf, 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 Tawaf arise.
Sheikh Ibn Uthaymeen held that proxy Tawaf (where the proxy goes alone without the sick person) is not established from the Sunnah and is a weak position. He strongly sunnah 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 Tawaf, as there is no time limit for Tawaf al-Ifadah. In extreme cases where recovery is not expected, some scholars from the Hanbali school permitted proxy Tawaf by analogy with proxy Haji.