Praise be to Allah.
It is permissible to do tawaf wearing shoes if they are clean. The Sunnah is to pray wearing shoes sometimes. If prayer wearing shoes is valid, then it is even more likely that doing tawaf and sa`i in shoes is also valid.
It is better not to do tawaf wearing shoes, so that people who are careless and who do not take care to avoid impurities will not follow your example, which would lead to the mosque being contaminated and made dirty.
The pilgrim may need to do tawaf and sa`i wearing shoes because of injuries to his feet and so on. So there is nothing wrong with him doing tawaf in his shoes , after making sure that they are clean. 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 Dawud (555) and classed as sahih by Shaykh al-Albani (may Allah have mercy on him.)
And Allah knows best.