A lot of the restaruants will have vegetarian options in Seattle. Except downtown. That is the only place I've ever been where I couldn't find anything to eat. Well, wait, there might be some with vegetarian options, but as a vegan I was **** out of luck in downtown. I'm talking like around Pioneer Square along 1st. New Orleans does have vegetarian stuff, though and the atmosphere was really awesome. I love that place.
If you have a car I really suggest going up to the University District. on NE University Way there is Sidecar for Pigs Peace which is a 100% vegan store and the money from it goes to Pigs Peace Sanctuary. Right next to it is Pizza Pi, which is a vegan pizza place. Across the street about 300ft down to the left of Sidecar and Pizza Pi when you come out of their doors is Wayward Cafe and they do vegan breakfast and lunch. Reminds me of a diner, omelettes, scrambles, country fried chiggin, things like that and then burgers and sandwiches and baskets of chiggin strips and whatnot for lunches. Every weekend they also have a special burger or sandwich they come up with.
Chaco Canyon is another place, they are on 50th right by the stuff I just listed above. They do raw and more healthy-ish foods than Wayward or Pizza Pi from what i've gathered but I haven't been there yet.
Down by the Seattle Center is a Chinese restaurant called Bamboo Garden that is 100% vegetarian and they are actually vegan except for their fortune cookies, the fortune cookies contain egg.
There is also Quickies, Sage Cafe, Plum Bistro, Thrive, Highline vegan bar, Georgetown Liquor Company, Arayas, etc. I really like going to Highline for a drink and hanging out. Tuesday nights they do Cake-a-roo-kee or something like that. It's basically kareokee with cake too. Pretty cool.
If you use happycow.net like someone above linked it can give you all the options. Oh, at the Pike Place market, across the street at the top there is a bakery on the corner that sells vegetarian and vegan baked goods. I've never had any, but it always looks and smells good when I walk by.
Don't worry, you will not starve as a vegetarian in Seattle. Even as a vegan you would not starve, not even close! Seattle is a very vegetarian friendly city and it's becoming a very vegan friendly city too.