If there is more than one person it is often cost effective to rent a car on Yakushima. There are many car rental outlets on the island - mostly near the airport and ports and once upon a time it used to be really tough to do it yourself if you didn't speak Japanese. The internet now makes it much easier and here is some details about how to find a good car rental company.
The first thing you might consider if you want to rent a car is whether your accommodation also offers car rental. More and more hotels and guesthouses are offering this service in English to their guests. Yakushima South Village, which is the southern youth hostel, is a good example. Morinokokage, which offers cabins near Anbo, is another.
Make sure you have a passport and an international driving licence. Without it you can't rent!
If your licence is from France, Belgium, Germany, Monaco, Slovenia, Switzerland and Taiwan you need to bring the licence and an official Japanese translation by the JAF (Japan Automobile Association) or by the embassy or consulate office in Japan.
You can now find many Yakushima car rental websites in English online. Here's a list of local and national rental companies:
Camgo (English website)
Carrentalgroup.com (English website)
Europcar (English website)
Toyota Rentals (English website)
Times Car rental (English website)
Nippon Rentacar (English website)
Orix Rentacar (English website)
For more information about Yakushima transport check out the one and only Yakumonkey Guide to Yakushima