Yes, there should be lots of places to rent a car around Mt. Fuji. I found the best way is to search car rentals near whatever train station you will be arriving. We rented a car in Tokyo and dropped it off at Mishima Station. So I guess you could also do the reverse. Kyoto Station to Mishima Station by bullet train and then go to Toyota Rental at Mishima and drive to Fuji Speedway. The drive is about 40 minutes.