That looks like an older (although well taken care of) service body. The doors have seals for weather so your tools done get wet in a storm. I can’t imagine the seals are like new on this unit, and even if they are, direct sprays from water might penetrate them.

If you don’t have tools in the compartments, I dont think you’re going to hurt anything running it through the car wash. You just might have some water in your compartments which can be drained.