I am vegetarian (I don't eat meat and fish). I have had dogs as pets in my life. More precisely, when I lived with my parents we had dogs. It doesn't really matter, but they were German shepards. My parents are not vegetarian, and the dogs were feeded with normal dog food https://en.wikipedia.org/wiki/Dog_food. I don't remember the type of meat this food had, but I assume chicken and similar things. It was ordinary good-quality dog food, sometimes of two types as suggested by the veterinarian.
Of course during our meals we would also give something to them (not the leftovers), and they basically tried everything we had that was not dangerous to them (all fruits and vegetables, meat, fish, pasta, etc.), in small amounts. This is to say that they were part of the family, and this is not a question about animal diet in general, or the ethic/moral problem of owning a pet if you're vegetarian/vegan.
A vegetarian doesn't eat meat and fish. More pragmatically, being vegetarian might be interpreted as a life choice that aims to stop the meat/fish mass production by not buying those products (this is the practical result that would be achieved if we were all vegetarian, whether one does it for ethical, environmental or other reasons) or buying them very rarely (this would have the same effect: you don't start producing a good if it's bought twice a year and it's not crazy expensive). Dogs are not vegetarian, and I don't want to consider vegan dog food right now - just pretend it doesn't exist. To survive, they must eat meat/fish. So the logical conclusion is that to live with a dog, you must buy meat/fish in form of dog food. This is in contradiction with your life choice.
The question is the following. To be coherent, do you think a vegetarian/vegan should never own a dog, or are there loopholes to my argument?
EDIT: some answers are focusing on the fact that dogs can be feeded with vegan food. I didn't want to consider this possibility simply because that's not possible for all pets. So if you like you can replace "dog" with "cat" everywhere in my question. I am interested in knowing if you think it's a coherent thing to do.