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