At the earliest stages, word of mouth is our best bet, IMO.
So who do we focus on telling about the site?
Veterinarians are an obvious target, but it can sometimes be hard to convince professionals to put time and effort into providing expert help for free, especially when it doesn't parlay into an increase in business for them. Only a specific subset of veterinarians will be likely to participate (those who enjoy such online communities, and for whom the inherent gamification of the SE platform holds appeal).
However, there are other audiences we can reach out to.
In particular, I know of one audience that is extremely dedicated, generally savvy with regards to using the Internet as a communications medium, and excellent at network building.
There are lots of people who use social media as a means of facilitating pet rescues. These people volunteer their time to work with local shelters, post rescue, foster, and adoption notices, and communicate with others doing the same to expand their geographic coverage.
I think our site may very well be of interest to them, as most seem to also be devoted pet owners. If we can engage them, and get them interested in the site, they may start sharing us as a resource across their already extensive networks. I know a couple that I will be contacting once this comes out of private beta.