My dog Daisy is a vegetarian. She would happily eat meat, but unfortunately it makes her violently ill and sends her liver into failure. So, although some on this board want to condemn vegetarian food for dogs, it keeps my dog alive. And she is thriving at almost 12 years old when meat had her one paw in the grave for the first five years of her life. Luckily, I threw my beliefs and preconceived notions to the wind, put my trust in a veterinary nutritionist, and vegetarian food it is.
If you want to feed your dog vegetarian food, then a good place to find one for $25 from a board certified vet nutritionist is here
Welcome to Pet Diets. I see no reason to avoid a vegetarian diet for your dog if that is what you want to feed him/her. Just make sure it is properly balanced. If I had my choice, however, I would opt for an egg-based diet first, followed by a meat-based diet, followed by a vegetarian based diet.
Good luck!