Rental Car Management: I work for a marine transportation company in the US. We use rental cars to move mariners to/from their place of work, which is constantly in motion, where they live for extended periods. They typically hand the car off to someone getting off the vessel for a return trip back home elsewhere in the US. We struggle to track who has what car. Is anyone aware of a system that is helpful in tracking a large fleet of rentals handing off between individuals?

i know this isn't a purpose built solution, but many project/portfolio management solutions come with a resource management add-on.  Your description could be described as assigning resources to a project (asset) for a specified period of time.  In addition to assigning people, you could also use a tool like this to assign milestones to the asset - like service and contract expiration dates.  Our company uses OnePlan for resource management and MS ProjectOnline for project/portfolio management.  There are other solutions like Planview, ServiceNow has something and probably many more.

If you can't find a premade solution, you could likely get someone to make a simple app (android/ios or web app) where your drivers could scan a vin to "take possession" of a vehicle, and use the app to do identity capture, and time logging.