Thank you for all the helpful replies about plants for Fire Island. Very much appreciated. I forgot to ask which vegetables would be suitable, bearing in mind what I have been advised about salt. Thank you.

Well kind of the same advice applies for veggies. But keep in mind that the animals might become more of an issue then you might want.
Everything that you will want to grow from lettuce to carrots, cucumbers, peppers, tomatoes are all tasty treats to squirrels, rabbits, mice and deer. You may want to consider fencing the garden!