my personal opinion on dogs
That is very interesting subject you have brought up.
As far as i know, dogs are primarily predatory carnivores that may exhibit scavenger behavior to survive. Because they are domesticated, they do not need to hunt (we feed them, so they don't need to worry), however, their predatory instincts are still there, and the games we play with dogs (such as "fetch" or having them chase us, or chase each other) are expressions of those very instincts. Moreover, their teeth are a clear indicator that they were not meant to chew plants, but rip apart chunks of flesh (i know it sounds awful, but carnivores are an important part of the food chain).
Moreover, a dog's main source of sustenance should be protein. They need way more protein than humans do, and i believe they also have different essential amino acids than humans.
Someone mentioned it here, and I have also heard of the claim that dogs would consume the content of the gut from the animals they'd be hunting. They would clearly eat the internal organs, but i find it hard to believe that they would go for the gut content of a herbivore. In all probability, the animals they'd bring down wouldn't even be on a full stomach, or they'd be at an advanced phase of digestion, so it seems like a stretched claim to me.
I would personally recommend you feed your dogs a meat based diet. However, if you do decide to go vegan on your dogs, make sure you find good vegetable sources of protein to incorporate the amino acids that dogs need. Good luck!