A small clarification: the need for a model release has nothing whatsoever to do with the photographer making money from selling the image. It has to do with commercial exploitation of the privacy rights of the model.
Lets say I take a great photo of Ichiro Suzuki at a game. I can sell that image to
Sports Illustrated for their cover for thousands of dollars without a release. A photo in a news periodical is considered
editorial use, not commercial use. My profit from the sale of that image is irrelevant. I cannot, on the other hand, sell that same image to General Mills to use on a Wheaties box. That would be
commercial use, and would require a model release. I could not even give General Mills that image for free, for "photo credit" (ahem!). That would still be commercial use, even if I make no profit at all. It is the
use of the image that determines if a model release is required, not my profit (or lack thereof).
This is why you normally don't require a model release if an event photographer is simply selling event photos back to the event guests. It isn't commercial use, therefore no release is required. Commercial in this legal context does not mean making money. Commercial means exploiting someone's likeness to promote a product or service.
Pets are a grey area in regards to a model or property release. As far as I understand it, a pet has no privacy rights. Only people have privacy rights. Therefore a pet doesn't warrant a model release. More likely, you'd need a property release if any release is required at all. It is a somewhat similar form for similar purposes.
Here is ASMPs tutorial and sample. They have sample releases for model and property, adult and minor.
http://asmp.org/tutorials/property-and-model-releases.html
All the legal issues aside, it is poor form to try to horn in on the business of the already-established event photographer. The event organizer has probably made special arrangements for that photographer to be there. Nobody will care if you take a few photos. But if you are clearly exploiting the event for your own profit and undermining the show's primary photographer, the organizers will likely eject you from the event. Dog shows are usually closed events, often on private property, and they have the right to boot you out for any reason. This is probably a more relevant consideration than the need for a model/property release.