The only mention of boots and shoes in the Lord of the Rings that also mention hobbits are mentioned here:
" There is in the text no mention of his acquiring of boots. There should be! It has dropped out somehow or other in the various revisions – the bootings occurred at Rivendell; and he was again bootless after leaving Rivendell on the way home. But since leathery soles, and well-brushed furry feet are a feature of essential hobbitness, he ought really to appear unbooted, except in special illustrations of episodes. "
So, it seems that no, they do not usually wear shoes and only appear on special occasions doing so.