Short Answer

When men are performing ṭawāf in the state of iḥrām, they will not be allowed to cover their feet anyways. However, women [or men who are performing ṭawāf without being in iḥrām] may technically wear socks or shoes. Remember, it is important to not make the area dirty, so only wear footing which is very clean.

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Shaykh Mustafa Umar