What part of the island are you planning on living?
Most restaurants are very capable of taking care of vegetarians, and even vegans. Because Thai and Japanese cuisine is EVERYWHERE, it's pretty easy to get by. Also, there's an all-vegetarian Grocery Store in Honolulu called Down to Earth. They have three locations on Oahu, I think. Their flagship store is near UHH Manoa, just outside of Waikiki.
The challenge with local cuisine is that it's VERY meat-centric. Hawaiians love their meat! Avoid local plate lunch places. Avoid local Drive-ins unless you know for a fact they have something you can eat (besides rice).
Check out the Happy Cow listings for Honolulu, you'll find a lot of places that will work for you. Because the general population here is somewhat healthy, and somewhat spiritual, vegetarianism is fairly common, and most places are willing to accommodate us.