I had to have a relative do the reservation for me. Was not cheap, and also had a hefty one-way surcharge when driving from Tainan and dropping it off in Taidong. Much more expensive than hire cars in Europe. But beats cross-island train travel, as the public transports for east-west travel are poor/time-consuming.
If you don't have a local licence to drive, also don't forget the international driving permit (IDP). They checked and photocopied when I picked up the car.