Re: "It was an ambulance that took an injured dog to a vet."
The structure of the sentence (a cleft sentence) is not very natural, however.
Probably your reader already heard something about "the" dog, and even though we may not mean a specific doctor, we tend to say "the" doctor, so (although I agree with Mike that the particular circumstances might make other articles more appropriate), probably "An ambulance took the injured dog to the vet."
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.