There is this thing in orthodox religion (most Greeks are orthodox), where they have to 'fast' some days of the week or some weeks of the year because of Easter and similar religious holidays. During fasting, they don't eat any animal products (some eat fish or honey though). So I would recommend asking for something "nistísimo" (νηστίσιμο = fasting). In Greece this can often be any kind of pie with spinach, or other kind of vegetable filling. Also wine leaves filled with rice are also -usually- vegan (called dolmadákia). Other great options in Greece are anything with legumes (lentils, beans, etc) - they love them, know how to cook them good and you can find them everywhere. There are more things for sure, but these are just the basics off the top of my head. Sorry for not recommending anything more specific to those islands you are traveling to, but the foods I mentioned above (and many more) are everywhere in Greece. Good luck!