It seems to me that a request for vegetables not cooked with any sort of meat or cheese product wouldn't be at all an unreasonable request.
a quick blanch in water, a just as quick saute in a pan with some oil, herbs and salt and pepper shouldn't take more than a few seconds (to properly cook the vegetables)
on the other hand, if you're eating in places that mass prep and overcook veg. and hold it in a steam table... that's an entirely different beast.
and if you're worrying about the health concerns for eating specific food products, you shouldn't be eating in those types of places anyhow.
point being:
any self respecting kitchen staff would be more than willing to cook veg. with no butter or meat, than cook a well done piece of meat for someone else.
It's simply quicker, easier, and brainless for a seasoned cook.
Working in restaurants is the service industry... and all about reasonable accommodation.
Busy or not, your request is simple enough that it shouldn't be a problem anywhere (and if it is, you might consider spending your money elsewhere)
All you'd have to do is ask your server.
"i don't eat meat or dairy, and would it be at all possible to have kitchen substitute some veg. cooked with just a little oil and seasoning?"
I'm not a vegetarian, but as i've mentioned before on here... I'm very sympathetic to dietary restrictions (by choice or not).
I've been lactose intolerant since i was a baby, and can't get cheese down to save my life... so i empathize.
i will say this: any food product prepared with meat, meat stocks, etc. is NOT vegetarian... even if you don't have physical pieces of meat in the mix.
...nor is gelatin (which in turn is extracted from bones in the stock making process).
Reading this thread makes me want to go eat at Zen Garden. (if you haven't... you should. It'll cover all your bases.

)
as far as downtown recommendations (and to support friends in the industry); I'd be willing to bet that Chef James Lucas and his staff at Bistro 301 wouldn't find any trouble accommodating you.