Yes! Under certain circumstances.
If the Command legitimately determines that the condition of the vehicle is a safety issue, then yes, the Command may order the Soldier to clean the car before operating it.
No different than a safety inspection prior to a long weekend or taking leave. If there is so much trash in the vehicle that some of it could slide under his feet while driving or a fire start from a cigarette ash, then absolutely. It all revolves around safety.
The Command cannot FORCE him to clean his car or punish him for not cleaning the car. However, they can order him not to operate or allow the car to be operated until it passes a safety inspection. Should the Soldier disobey the order, then the Soldier can be punished for disobeying an order.