This is a matter of scholarly discussion. The majority of scholars from all four schools prefer that the sick الحاج be physically present for الطواف, even if they need to be carried, pushed in a wheelchair, or transported on a stretcher. الطواف is a physical act of worship closely tied to being physically present at the الكعبة, and the scholars prioritize the الحاج's presence.
الشيخ ابن باز stated that a sick الحاج should be carried for الطواف if possible — either on someone's shoulders, in a wheelchair, or on a bed/stretcher. He cited the report that النبي performed الطواف on his riding animal during the Farewell الحج. If the الحاج is unconscious, their family should carry them for الطواف, 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 الطواف arise.
الشيخ ابن عثيمين held that proxy الطواف (where the proxy goes alone without the sick person) is not أُسس from the السنة and is a weak position. He strongly مستحب 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 المستشفى and cannot be moved at all, they should wait until they recover to perform their الطواف, as there is no time limit for الطواف al-Ifadah. In extreme cases where recovery is not expected, some scholars from the Hanbali school permitted proxy الطواف by analogy with proxy الحج.