Dogs are actually omnivores. I don't know if they could subsist on a vegetables and grains alone any better than we can, but their diet definately doens't have to be all meat, which cheapens it.
I know, but at the time these chariots may be developed I don't think the people would think to feed the dogs plant materials. While the dogs may eat vegetables and such on their own, the handlers would look for meat for them.
If we need an omnivore to pull the chariot, maybe these people should develop a warpig (warboar?).
Pigs are intelligent, and while not as useful as a horse or dog in combat, breeding those traits best suited for war would not be too difficult. While not often used as work animal, there is no reason they could not be bred to be so.
Say it starts with someone getting the idea to use pigs to help plow a plot of land. As this catches on, some people start to use the pigs to pull carts with produce and goods. Eventually pig pulled cart races become popular. Strong , faster, and controllable pigs are bred. Food is readily available and the pigs breed fairly quick compaired to cattle. The downside to using the pig is the number of diseases they harbor, their lack of sweat glands (while canines pant to keep cool, pigs need water or mud), and skin sensitivity to sunburn.
Dogs cannot attain the size or build necessary for pulling wheeled vehicles. Our bigger breeds have a number of health problems without being put under that strain.
But they could if bred properly.
Back when I was in high school we would have people come to demonstrate cart racing with various animals pulling (I went to a Vocational Agriculture school BTW). One of the races had some people with large dogs.
While this isn't the group I saw back in school, it is a real sport.
http://members.aol.com/_ht_a/rottweilerdriver/DogCarting/index.html
And from wiki
http://en.wikipedia.org/wiki/Carting