Tried at China Trust (Zhongli branch) and they said I needed a 2 or 3 year ARC (?!). There wasn't anyone there who could speak English though, so had to ask a local friend to call back later to check that there wasn't a misunderstanding.
Then tried at Taishin (also Zhongli). They were friendly and spoke enough English to talk me through the Chinese forms (their requirements: ARC, health card, and passport). Got the card (though not sure exactly which one) in around 30 minutes. I'll try it out online later...
Update: Was able to order a book (didn't use Amazon) and book a hotel (Agoda) with the card. Great success! Thanks for the thread Yuli (:



{/anecdote}
I was able to get a Visa Debit Card at Bank of Taiwan with only a visitor visa and the paper with the ID number from immigration office 



