Your dog is Vegetarian?
So that would be because
a) After long and serious discussion, your dog decided that it wasn't happy with the ethics and morality of eating another creature. In its new Dog Charter it also agreed not to chase cats, worry sheep, or foul public footpaths
b) You are denying your dog the diet to which it is best fitted to digest, so that you can inflict your, carefully considered, and a little warped, views, onto that of another, non sentient being.
If you don't believe me, offer your dog two bowls tomorrow, one vegetarian and one with some raw beef steak. I'd lay a bet I know which bowl it would go for.
On a similar note, hunting foxes with hounds has to be amongst the most truly natural ways of killing a fox. After all dogs hunted other animals in packs long before domestication.