I have actually seen quite a few variations on this, car dealers were a big customer base for VIdeoIQ, and so I've been to a LOT of dealerships.
Primarily, you want a camera that you can lock the shutter speed to a minimum and still get a good exposure. This mostly means controlling the light around the camera, and ensuring things like headlights/taillights do not cause exposure issues, or cause the camera to start adjusting iris, shutter speed, etc. A WDR camera would probably be a good idea for this application.
The best approaches I saw were cameras inside the service bay on pull-through arrangements where the service writer desks are inside a garage area and the customer pulls in and stops to drop off their vehicle. In these cases the lighting in the garage/service area is usually pretty consistent, and bright enough that headlights/taillights do not cause issues. (It sounds like this is the case you have). Plus, the vehicle is stopped, which helps get a clear image.
If you are going to place the cameras outside to scan vehicles pulling into an area then you may find that adding additional illuminators, even during the day, will help give you even and consistent lighting, or least a minimum guaranteed light level.
If your VMS supports it, a POS integration with the service-ticket system could be helpful to tag or bookmark video with the service ticket in case you did need to look it up later.
For the under-vehicle part, there are companies with products like this, you often find them from the vehicle barrier companies, as they are often used for customs/border and similar high security applications. I do not have any specific product recommendations there though. It is basically a camera or two mounted inside of a speed bump.
