I would think it was easier for a vegetarian to be on LID. But I'm not a vegetarian, so I guess that assumption was wrong!
I just finished my LID and had my RAI treatment this past Monday, so the misery of "what to eat" is still fresh for me. I tend not to eat much meat anyway, so I just took a more vegetarian approach. I ate lots of pasta with salt-free/iodine free seasonings and olive oil. Oatmeal with sugar and raisins. But I did read one guideline that said no dried fruits.. The raisins didn't seem to cause me any problems, though. I snacked on bananas and applesauce, as well as other fruit. Unsalted matzohs with jelly (read the ingredients for dyes and other no-no's.) It was truly depressing to walk through a grocery store, read the labels, and walk out with almost nothing to eat! The first two or three days were miserable, though. It seemed like I was hungry all the time. It got better.
Have you been to thyca.org to check out the Low Iodine Cookbook? There might be some ideas there to spur your imagination a bit.
Also, if you are preparing for RAI, look for Atomic Warhead Sour Spray to use after your treatment. I found this worked very well without giving you a sugar overload (or cavities!).