There are always zombie animals, it's just they are often overshadowed by human zombies because they largely outnumber animals, especially in rural settings, and most animals instinctively run away from zombies rather than try to fight them head on, like most humans do.
The most interesting and terrifying realization about animals in a zombie apocalypse can be best illustrated in the third Resident Evil movie, Extinction. With that massive outbreak of zombies polluting the natural environment, many of its carnivorous animals either die out or become extinct because of the lack of herbivores to eat or being unable to hunt in the changing terrains. The herbivores that do survive, which are usually scavengers, are forced to feed off the flesh of "killed" zombies that liter the wasteland, in turn, contracting the virus and spreading it to other animals that they attack.
In the movie, Alice, the protagonist, witnesses a crow feasting on the flesh of a recently "killed" zombie. A flock of crows, or a murder, to be precise, then proceed to attack a caravan of survivors, who bravely fight them off with a mounted flamethrower, and later dispatched by Alice.
So yeah, even if humans are able to find a cure/vaccine to counteract that virus, there is still a possibility that animals can contract the virus and have it change overtime (like the flu or common cold) that it will still pose as a threat in the future.
