Sorry, 'Carrier' was his occupation as stated on the census.
No, I do not know that anyone came from Ireland, so far the Webbs that I have found are firmly rooted in Romsey. They are not moving around, and they are not missing from censuses. They are living in houses and doing a variety of occupations.
Even if they did come from Ireland, that does not automatically mean they are travellers.
If your family think their ancestors were travellers, do they mean the Webb family, or one of the wife's family were travellers?