We used to live in Vero, and found the Patio to be reliably our favorite restaurant. It's not much to look at from the outside (OK, it's ugly), but the service is always accomidating and the food dependable.
The Ocean Grill is just fine, too. Re the "looks" from other patrons: Vero is a rich man's town in a lot of ways, and there are a lot of folks who move down there to escape "whatever." They may give looks to kids, but they probably give the same looks to most anyone else who annoys them. (If that sounds like a biased statement, it is!)
BUT -- my #1 call for a restaurant that is not too much of a drive is Dixie Crossroads in Titusville. Kid friendly, a local favorite, a waypoint for folks traveling the coast, and I PROMISE you won't be dissapointed. VERY reasonable. Their claim to fame is their "discovery" of rock shrimp. They split 'em, butterfly them in their hard little shells, toss 'em on the grill with lots of butter, and the results are absolutely delicious. You'll find no other shellfish from shrimp to lobster that tastes any better. There's a playground for the kids, fish for them to feed, and you'll have a great time. Get there EARLY; I can't remember if they take reservations or not. It is worth the trip, especially if you can link it with an excursion to Kennedy Space Center.