Finally got it! The online application really does work. I applied online and got an e-mail response telling me that my card will be available after 3 days at the Unionbank branch I specified. I went there after 8 days so I can’t really tell how fast they are but when I got there my card was waiting for me. Paid the 350 peso annual fee and that’s it!
In case you’re interested, here’s the link to the Unionbank E-On VISA Electron information pageonline application form. Now I’m one more step closer to having my Paypal account activated! Will try this over the next few days.