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