That relationship would blow up on the launch pad with me. Everything I cook has meat in it, everything I like to eat has meat in it (except for my breakfast cereal---they haven't perfected Bacon Krispies yet), and I'd very likely starve to death if all I had to eat was vegetables and grains (even if milk, eggs, and cheese were allowed I'd still have a sharp reduction in my rations).
Now, if I had a coffee shop-casual fling with a vegetarian (as in, talk over coffee, go back to my place, fuck until satisfied)? That's different. But a full-fledged relationship with a future---no. Wouldn't work out.