Below is the web-site of the on-line Taiwan visa application.You fill in the form and summit, then you print out the confirmmation,(make sure the print out can see the confirmmation number),then you can choose to pick the visa up at the HK airport(counter K), be sure you have to pick it up before 8pm, because the counter open from 8am to 8pm daily. Normally, it takes 5-10 minutes.Very convenient.
https://nas.immigration.gov.tw/nasf/ctlr?PRO=PRO_Task01Application
When you arrive at the airport, go to counter K ,give the passport, ID card and 55HKD to the staff and he will give you a number card, when you see your number appear on the screen, you can go to pick up your visa. One person,one application.
You also can go to中華旅行社 to pick up the visa. The procedure is : go to the counter, give the print out and the passport to the staff, wait for 20 - 30 minutes, the staff will call out your name,then give her money and pick up the visa. May be you can call 中華旅行社(25258316) to ask for detail.
Apply the visa at Taiwan airport:Fill in the form at the counter,give them your ID card and 200台幣, wait for 10 -15 minutes, depends on how many people apply the visa.
PS: The visa that apply in HK(Both at the airport and 中華旅行社) is valid for 2 times entry, each time for 14 days;
The visa apply at Taiwan airport is only for 1 time entry (also 14 days).
參考: my experience, I tried all 3 methods and pick up at the HK airport is the most convenient.