I thought I should follow up and share what I learned from this process.
I was finally able to get the transfer to work from Vanguard to our bank in Taiwan. All that was required was the SWIFT number for the New York branch of our Taiwanese bank. Fortunately, this bank does have a US presence.
* Vanguard's documentation for international wire transfers, including the form itself, lacks clear instructions. This is mostly why I was confused.
* Complicating this was the language barrier. I don't speak Chinese so my wife had to get the information from the Taiwan banks (both in Taiwan and in NY, since the rep I spoke with there had a heavy accent). And since I manage our finances here in the US, my wife probably wasn't completely conversant in just what Vanguard needed.
The bottom line is that this has been resolved and we successfully made a wire transfer from our Vanguard account to my wife's bank account in Taiwan. Too bad the dollar is in decline.
Many thanks to all who responded with tips. I am grateful.