The website also sounds like every foreigner can just open an account… 
But on that topic: My last transfers using E.SUN all went through without having to go to the branch. All around NT$ 300k - I just had to use the right remittance code: Remittance code - which one to choose?
I had to make sure, though, that my sending account uses my Taiwanese address and that I specify the benefactor name in the form “LAST NAME FIRST NAME”.