YES, if you purchased comprehensive coverage, which is no fault and hitting an animal is hard not to do
With an animal hitting a car is easy to prove, since the animal will leave either blood or hairs from the impact. But the biggest problem of hitting or (not) hitting an animal, is that you do not hit them, thus end up hitting another car or object (tree) is an example. So, if able to avoid hitting an animal and hit something else, then it is collision (fault) and insurance will still pay, (if) you bought collision coverage.
In most cases, a police report is not necessary, but a good idea. Either way, the blood/hairs is the proof of hitting an animal.
retired auto adjuster