The problem is, Greeks don't really 'get' why anyone would want to be vegetarian ( - or vegan).
As far as they're concerned, humans eat meat and it's unnatural not to, lol.
(I'm afraid, I have to say I agree)
Having said that, as I'm sure you know, the restaurants always go out of their way to please their customers, so I think if your family are specific in their requests, they won't have a problem. Plus, given the number of 'happy clappies' now in Arillas ( - many of whom I'm sure have similar requirements), the restaurants are probably fairly used such requests.