A vegan diet can be just as healthy (or unhealthy!) as a diet that includes meat. Since I am not a veterinarian, I am unable to answer your question about a vegan diet for dogs. I recommend talking to your vet about that.
Keep in mind that a vegan diet requires careful tracking of your intake of protein, fats, carbohydrates, and vitamins and minerals. By not eating animal products, you may not get some nutrients, like B vitamins or iron. These nutrients are essential for our bodies to function properly and this means you might need to take supplements.