Asheville with kids
Let's see, here are a few to get you started. These are all local/regional, casual kid-friendly places that shouldn't have too much of a wait (if at all):
Mela - lunch buffet
White Duck Taco
Grab picnic items from Greenlife or EarthFare to take to Carrier/French Broad parks - walking trails along the river connect the two
Picnic/Grill at The Bywater - bar on the river with lots of rooms for kids to run around and play games
Wicked Weed and LAB are brewpubs with decent food - always kids at those it seems during the day. Pack's Tavern is OK too I guess and is right next to "Splashville" - a water fountain feature on the square downtown where kids are always playing in the water.
We do Asheville for a few days every summer with a now 11 yo. Our two "can't misses" are Curate and 12 Bones. 12 Bones may not be what you're looking for, as you asked for no BBQ and you have a vegetarian, but this isn't traditional NC cue and my kid thinks the mac and cheese is the absolute best she's ever had. The corn pudding is outstanding, too.
We look forward to Curate every year. Check the prices first, though, as the tab can run up quickly.
We've done LAB and felt kinda neutral about it. No problem bringing kids, though.
We've also liked Early Girl Cafe and Doc Chey's. Their kids's menu is actually half-sized portions of the adult menu. I'd skip the Mellow Mushroom. We have a few at home and odds are you have better pizza at home, too.
This year, we're trying to decide whether to get to Admiral or Seven Sows.
We love Asheville and go often. We do not have kids. Some friends of ours went there to tour the breweries for a few days and had a hard time with their children, but once again, they were going to drinking establishments. Let's face it, if the main purpose of a place is alcohol, it isn't meant for kids.
Just about every place in Asheville has vegetarian choices. Some of our favorite dining choices that would be good for kids include 12 Bones (cheesy grits and collards are awesome in addition to the BBQ and ribs), BiscuitHead is indeed wonderful with vegetarian options, Salsas portions are huge so could easily be split between an adult and child, Barley's is a beer and pizza joint where you can each order your own toppings on a slice by slice basis, Laughing Seed is entirely vegetarian and very good with a patio, Wicked Weed is awesome with great food and could be considered kid friendly if you sit on the patio.
A lot depends on your kids. Many of the restaurants in/near downtown are pretty small and if kids can't sit through a meal they would disrupt everyone. Most of them do to go orders, and Pack Square is a great place to picnic.
Thanks everyone. We went to a number of suggested places. Chai Pani and White Duck Taco were our favorites and we also had a great "breakfast for dinner" one night at Early Girl. I also would suggest The Chocolate Lounge for amazing dessert and the Double Decker Bus just for fun (not for the food or coffee). Wish we could have made it to the Wicked Weed or Curate, but maybe on our next trip when they are a bit older. Thanks again.