In an act of the purest optimism I called their tech support line. A really friendly sounding woman named Mary answered the phone and explained that the reason I could not attach files was because I was clicking on the "Attach button." No that's not a typo. Keep reading.
The conversation went something like this - I rattled off a bunch of info: Browser version, OS, Steps to reproduce the problem in multiple browsers. I was a veritable fountain of debugging information. She listened patiently and when I finished...
Mary: "Have you tried clicking a half inch to the right of the attach button?"
Me: [stunned]: "That honestly did not occur to me."
To my pure astonishment it worked. I've used a computer 5 days a week for almost 11 years and never seen anything quite like this bug.
Thanks for the help Mary! We only spoke for a minute, but you'll be long remembered as having provided the weirdest working technical solution I've ever heard.