You can charter a car online, you can find it on travel websites, or you can find it in Taiwan. Many taxis in Taiwan are electronically tuned, so you only come by phone. Hotels will have a car dealership phone number, just ask them.
Do you know of any formal channels for chartered cars in Taiwan? You can look for it on Taobao, find the one with high credit rating and large quantity, and book it on Ctrip if you don’t want to worry about it. Ctrip is considered a formal channel. <br/>How to charge? Chartered cars in Taiwan are based on the day (charged for 8 hours a day), overtime plus money, as well as the number of people (there are 5, 7, 9 and more models) and routes (different prices for different routes).