The cruise ships aren't nearly as good as WDW when it comes to food issues, including a vegan diet. I am a vegetarian with a dairy allergy so I'm almost a vegan (and don't eat eggs on their own or honey). My suggestions are as follows:
Talk to your server the first night and explain your dietary needs to him/her. Be very specific - no dairy, no eggs, no meat, no honey, etc. If you don't feel that person understands, ask to speak to your head server.
Be mindful of the bread course - due to my allergy, one evening, I asked because I thought something might contain dairy, and sure enough it did. They brought me some other rolls that were dairy free and each night after that, they brought the same rolls.
For the vegetarians, be very cautious of any dishes that contain cheese - most likely, none of them are vegetarian cheeses. Especially the dishes with parmesan cheese - no parm is vegetarian unless it is "fake" parm.
Most of the veggie entrees were very small - they don't seem to think we want a separate veggie and starch in addition to our entree. Feel free to order extra or a side of something you might like.
The chefs are very happy to concoct something for you - just ask the night before to order. My best meal on the Dream was when I asked if the chef could create a vegan meal for me based on Indian cuisine (many of the chefs seem to have an Indian background). I got a fabulous meal with tofu kabobs and Indian curry veggies that was to die for. And, he made me a really nice vegan dessert too. He came out to see me and was very happy to have been able to make me something special. He was disappointed that we weren't eating in the dining room the following night - we were in one of the Royal Suites so were having Palo in our room that night (if you do Palo they will do some wonderful dishes for you!).
The menus differ on the "special" nights for each restaurant. The Pirate night menu was the worst for vegans/vegetarians - the veggie meal was 3 vey small sausage looking things in a mushroom sauce - to be honest, they looked like turds. And, the only decent vegan appetizer option was a cold soup that just didn't add up. I left feeling very hungry.
I think my best advice is to be very proactive and vocal about your wants and needs. The chefs are happy to create within the limits of what is on the ship. Also, be understanding that many people won't know what a vegan diet is - they won't understand the no honey issue, they think all cheese is vegetarian, and they certainly won't have vegan sugar on board. To the extent you can overlook things for a few days, you might have a better cruise. I know that when we travel to the Caribbean or South America, I always assume the beans are vegetarian, even though I know they probably aren't....