Meat – Grilling veggies & meat on same surface for vegetarian guests

My response to this kind of question is always just ask, and if you absolutely can't, err on the side of caution.

I'm assuming here that you're talking about a pretty thorough heating and brushing. If you're leaving a bunch of meat stuff on the grill, that someone could conceivably taste, that's not good - you certainly shouldn't be risking food that actually has meat in it. But if it's essentially clean, with pretty much everything burned and scraped off, you're in a much better state.

I believe most of the vegetarians I know would be okay with that general plan. They're pretty pragmatic about their diets, they they know that some people eat meat and aren't going to have two grills, and they'll regard it as pretty similar to you using a pan that previously was used for meat but has since been washed. I would be sure I was doing everything that I could, thorough heating and brushing, separate (or washed) utensils, and so on, but I wouldn't be overly worried at that point.

But there are also people who keep to much stricter diets for all kinds of reasons (see Journeyman Geek's answer, for example), who would not be okay with this.

The only way you can tell the difference between the two is to ask, or to know someone well enough that you don't have to ask. This is really a pretty general principle when it comes to potentially strongly-held beliefs: if you want to do right by someone, don't expect to be able to use generalized categories to get the answer, assume you need their answer.