Some of my favorites are:
At DL:
The Three-Cheese Monte Cristo at Cafe Orleans. It's a bit heavy, but I usually share it with DF.
The Fried Green Tomato Sandwich at Hungry Bear.
At DCA:
Carthay Circle often doesn't have much on the menu for vegetarians, but they will make you something special if you ask. Every time I have, it's been the best thing I ate that day.
Wine Country Trattoria offers a few different vegetarian pasta dishes, and I've never had a bad one.
The vegetarian shepard's pie at Flo's is pretty good, but the portion is small, and some people think it's too bland. I like it.
Outside the parks:
Uva Bar's menu changes frequently, but they always have some good vegetarian options. Get the dip trio, if they have it. SO GOOD.
The Crispy Tofu Salad at Tangaroa Terrace. (At DLH)
Steakhouse 55 (DHL) has the best breakfast, IMO, and aside from a few already veg options, they will also modify their other offerings. They have one of the best eggs benedict I've ever had.
EDIT: I forgot to mention - beware the desserts. Almost anything with a mousse or frosting has gelatin in it, including the yummy cupcakes.