Our renter let her incontinent dog urinate in three carpeted rooms in the house down to the wood floors. Is this vandalism?

You can sue her for damages.

If the security deposit is not enough to repair the damage you can take her to small claims.

This is most likely not vandalism as you have to prove that she intended to damage property. You can however sue her for the damages you sustained.

Unfortunately, while this probably seems to be vandalism, it will not likely be actionable. Instead, it is more likely to be treated as an act of waste and damage to the property. If the tenant had a deposit, I would think this is more than ordinary wear and tear which would be covered by the deposit. I suspect that the deposit will not be enough to replace the carpet and treat the urine spots. You will need to sue the tenant for damages and breach of the rental agreement if you want to recover the damages. You should consult with an attorney to see if this is something that would be worth pursuing.

It is not vandalism unless it was intentional, get an attorney if she is collectable for the rent and damages due.

No, but it should be covered by your insurance.