I brought my two wonderful cats over with me early this year when I moved. I went to a random vet in Zhonghe who was a nice guy, but I'm sure most vets can help you out. Ask what you need to do immunization-wise to ensure they can be taken out of the country. There's a list at the 防疫檢疫局 website that I can't seem to find right now which you should check against. Your vet will 99.9% likely issue you an all-clear in Chinese only.
If you're in Taipei, you're in luck -- the 防疫檢疫局 station at Songshan Airport is easy to get to. They will type up an English-Chinese bilingual report that you present to your airline when checking in and again to the Customs officials at your destination. If memory serves, there is absolutely no charge. See here for the location: https://www.baphiq.gov.tw/content_edit. ... ypeid=1298
And if you're not in Taipei, they also have offices in Bali (New Taipei), Hsinchu, Taichung, and Kaohsiung.
I'll help with any other questions I can.