We were on the Nieuw Statendam for a week this past spring. I'm a vegetarian (but I eat dairy) and I have an allergy. We typically ate dinner and breakfast in Club Orange.
There was a separate vegetarian menu with items I could order anytime during the week, but I had to insist a little before staff admitted it existed. (Initially I was told to just order from the main menu.) I made the mistake of giving it back the first time I had it, without taking pictures.
There was also typically one vegetarian option per category on the main menu. It changed daily.
I ordered dinner the night before. I was allowed to see the next day's menu in case I wanted something from it, otherwise I ordered something on the vegetarian menu. After the first night, that worked well.
There was a vegetarian soup and a vegetarian entree in the Pinnacle Grill and I enjoyed both.The soup had dairy, not sure about the entree.
I didn't find the buffet to be as convenient, because it wasn't always clear to me what (if anything) would be available that I could eat before standing in line at a particular station. I did get "New York" pizza a few times, which I enjoyed, and I would guess they could probably make one without cheese.