If you have hit an animal, such as a deer, on the roadway then the claim would be placed under the Comprehensive portion of your Physical Damage coverages. How long you can wait to make a claim for the damages sustained due to hitting an animal will depend upon state statute of limitations for filing claims as well as the terms of your policy.
Typically your insurance policy will state how long you have to make claims under your the various portions of your policy such as Comprehensive. The closer to the actual date of the incident that you make the claim with your insurance carrier the better. Also if you have a police report an insurance company would normally want a copy to see that you did hit an animal and that is how your vehicle sustained damage.
Since state laws vary on how long one can take to make a claim (or file a lawsuit). If there is no statute of limitations that a state mandates then an insurance company has their own internal guidelines that one must follow to place a claim within a prescribed time period. To give you a very generalized time limit for making claims it would normally be required to be done within a year or 2 for physical damages to a vehicle.
There are some states that have specific laws or statutes with regards to filing claims while others may allow your insurance policy and insurers guidelines dictate how long you have to place a claim. This is also true of how long an insurer has to respond to a claim. Some states have specific laws in place while others just say within a reasonable amount of time.
Most all insurance contracts contain provisions stating how long you have to file a claim or suit against the company. This is known as a Contractual statute of limitations. If you do not file a claim or a lawsuit within this time period, you may lose all rights to recover for your claim.
To find out how long you have to make either a Collision or Comprehensive claim with your insurance provider you can read through your policy and speak to your insurance agent. To find out if there are state laws that limit the amount of time you have to make this type of claim you can contact your state's insurance regulator whose consumer division should be able to give you information on your situation.