Hello Ahara!
Technically, dogs can be vegetarians. For example, the President of Slovenia had a dog who ate vegetarian food, though I'm not sure if by his own preference. He's dead now. I have a dog as well and I've never thought of feeding him exclusively vegetarian food. Mother Nature created dogs as beasts and as such, they are mostly carnivorous. You don't need a special veterinarian knowledge to know that. Dogs teeth are shaped to tear and chop meat. They can't chew food like humans, and vegetarian food needs to be chewed because it's digested much slower than meat. Beasts' intestines are shorter than those of herbivorous animals (including human). Because of that dogs can't digest vegetarian food efficiently.
Then there's another thing: I believe you have to take the law of karma into consideration. If a being is reborn as a beast (dog), then Mother Nature has probably meant it to be carnivorous.