The short answer is that dogs can be vegetarian, but they do not want to be.
Unlike cats, who are what is called obligate carnivores, while dogs primarily eat meat, they are opportunistic omnivores. They can survive without meat, but they cannot thrive without meat. Dogs’ digestive systems process meat protein most efficiently. They can digest some plant matter, but not very efficiently, and they do not use plant protein as well as they do meat.
A balanced dog diet will include some grains, vegetables, and fruits but will include a large proportion of animal-based protein. It is best to avoid dog foods that contain a lot of fillers and foods that many dogs don’t digest well — corn and wheat are problematic for many dogs. Symptoms of poor digestion include gas, bloating, and loose stool.
I do understand the problem of vegetarians who do not want to have meat in the house. I am a vegetarian surrounded by omnivores myself. If you are adamant about keeping a vegetarian home, perhaps a rabbit or a goat would be a better pet? Or a miniature horse?