A claim with your insurance company will usually cover any damages your car suffered while getting it washed. As long as you don't sign a subrogation waiver, your insurance company is free to try and recoup their losses through the subrogation process.
Subrogation describes the process in which an insurance company attempts to recover losses from a third party that caused the damages or loss incurred by the insured. Since this process takes place entirely between insurance companies, policyholders aren't usually involved. You can however, always discuss with your agent if they believe it's prudent to pursue a subrogation claim, as it can be different on a case by case basis.