Since so many carnivores like to cook their meat outdoors on a grill, they think it is the only possibility. Vegans cooking out cause even more consternation than the fish issue (you know that thing where they can't understand that fish are animals).
It is advisable to have your own grill. A cheap 20. dollar one can be had. This prevents your food from getting meat juices and ashes on it. Call it the Virgin Vegan grill : never had meat on it, never will.
Corn on the cob works well to grill. Just throw it on. Also try shishkabobs with all manner of vegetable goodies on it. I like pineapples, mushrooms, baby corn, broccoli, and cauliflower. Possibilities are endless here. The large sliced portabella mushrooms will fry just as a hamburger will, and make excellent sandwiches.
Also use foil. Aluminum foil made into a bowl shape works well to cook mixed vegetables on a grill. Or wrap potatoes with it and place on top for awhile.
Pre-thawed frozen burritos are also good when cooked on grills. Another great food is fried mush(corn meal). Just make some up and refrigerate overnight. Then cut into slices- normally to fry in a pan, or put on the grill.
Use your imagination. Also see this article for some ideas.