I personally think it wrong to impose a diet such has vegetarianism on an animal that simply cannot say no either way. More so on those who do only because they are vegetarians themselves.
While it may be true that dogs in the wild are known omnivores, they also require nutrients found In meat such as taurine & while some vegetarian pet foots may contain taurine supplements
The truly beneficial ones come with a premium price tag, so I would assume you will be paying these to ensure the health of your pet & not simply theming a lettuce in the bowl.
Also, in response to Boaz first post, we domesticated dogs over 10,000 years ago. Since then, we have controlled how they developed and their diet. So really, me feeding her a vegetarian diet is okay