A bit outside the city, but Fest (Biergarten) is very family and dog friendly. Love the food there, and though there is beer, I don't drink and don't feel out of place there 🙂

Also in the next shopping center over is Vampire Penguin, a shaved snow place that is also dog friendly! They do pup cups for dogs and their shaved snow is one of my favorite desserts in the richmond area.