Hi strawbie on tour,
It really depends when you want to visit Hokkaido. If you are visiting in the winter months, I personally do not suggest driving, especially if don't have experience with driving on snow. Hokkaido gets the biggest amount of snow in Japan (virtually snowing every day), most of the time the roads are not properly snow cleared, 99% of the time they are icy and tricky to drive. In Hokkaido people don't grid or salt the roads, which it makes more difficult to drive. As it's quiten windy island quite often you can get snowdrifts on the roads too.
All car rent company will require an international driving license and in the winter months it's compulsary to rent a 4x4 car for which you have to pay extra.
Toyota rent a car and Nippon rent a car are one of the most popular rent a car companies in Hokkaido. Both have desk at Chitose airport in the ski resorts around Hokkaido and Sapporo city.
If you are going to stay at one place while in Hokkaido, I personally recommend using the train or the bus companies (Big runs, Resort Liner, Chuo, Donan, Whiteliner) as they have reqular services to almost every where in Hokkaido.
I hope that helps. about 5 years ago