yes there are farmers markets, though they are known as country markets here.
san antonio has one quite near to the sunset strip, and the ibiza town one is down a backstreet and it would take me hours to explain how to get there.
however if you want fresh local produce the main markets in san an, ibiza and santa eulalia are brim full of mostly local produce in the fruit and veg category.
fish are usually clearly marked as being caught locally, although the opportunities to purchase local meat, mostly lamb and pork, are rare and seasonal. The country chickens (they're a yellowy colour because they eat real food and not quite so tender becasue they can run around) are abundant all year.
having said that the artisanal centre can caus near santa gertrudis sells local meat, both in its shop and restaurant, and actually farms lambs, pigs and goats.
don't forget that wine is produced in ibiza and the two largest producers at can rich near san an and sa cova near san mateu allow visitors to their vineyards. otherwise the wineshop enoticum has a wide selection of local wines.
have fun
8)