What are my food supply options sailing the Cyclades?

We are going on a two week sailing trip through the Cyclades, and I was told that we have to stow all the foodstuff we will need for those two weeks on depature, at Lavrio, because "there are no shops at the Cyclades islands". While this is not a problem with drinks, oil or potatoes, I feel uncomfortable with weeks-old bread, fruit, vegetables, or fish. What are my options?

First, is it true that there are no shops at the Cyclades islands? I almost cannot believe this. There are townships on these islands, there are people, there are harbours, there must be some regular food supply. However, at least Google Maps does not show any hits for παντοπωλείο (grocery).

What about markets? Is there a way to obtain a list when and where markets take place in the region, or is there a rule of thumb?

Any other suggestions? Do I have to stow a fishing rod, and some flour and yeast and bake my own bread on board (given the caboose has an oven)?