It is permissible to do tawaaf wearing shoes if they are clean. The Sunnah is to pray wearing shoe sometimes. If prayer wearing shoes is valid, then it is even more likely that doing tawaaf and sa‘i in shoes is also valid. For more information please see the answer to question no. 69793.
It is better not to do tawaaf wearing shoes, so that people who are careless and who do not take care to avoid najaasah (impurities) will not follow your example, which would lead to the mosque being contaminated and made dirty.
The pilgrim may need to do tawaaf and sa‘i wearing shoes because of injuries to his feet and so on. So there is nothing wrong with him doing tawaaf in his shoes, after making sure that they are clean, because the Prophet (blessing and peace of Allah be upon him) said: “When one of you comes to the mosque, if he sees some dirt on his shoes, let him wipe them and pray in them.”
Narrated by Abu Dawood (555) and classed as saheeh by Shaykh al-Albaani (may Allah have mercy on him).
And Allah knows best.