Vinegar is a weak acid so it will take a long time to dissolve the Calcium deposits. Try soaking the pots in straight (5%) vinegar if you want to use it. Alternately, buy some muriatic acid (pool supply) which would remove the Ca deposits probably instantly. High fired clay pots are not porous, clay vitrifies at high temperature - that's why high-fired clay pots can withstand freezing temps - because there is not water in the non-existent pores to expand and crack the pot. If you're concerned about the acid hanging around then use baking soda (Calcium Carbonate) to neutralize it.
All that said - I would do something much more simple - coat the pot in walnut oil (from mashing walnuts in an cloth sock, not from a commercial can of oil). This doesn't remove the deposits but it does make them hardly noticeable. The oil will wear off after a while but even if you clean the pots with acid the deposits will return anyway.
Or switch to using RO water, that will eliminate the problem and make your plants grow better all at the same time. Basic RO systems start around $150 - look for one at a hydroponics supply - they're designed for high volume while not filtering quite as much of the dissolved salts as the drinking water type and they waste less water to make the RO water.