Hey guys & gals, I am a Steam user having this issue and I figured out how to get it to work.
Here's how:
1) Go to elderscrollsonline.com and select "Sign in through Steam"
2) Go to your account info and write it down/memorize how your name and email are entered
3) Log out and reset your password
4) There may be a back and forth with a code being set to your email
5) Now, launch the eso64.exe game client directly (e.g. C:\Steam\SteamApps\common\Zenimax Online\The Elder Scrolls Online\game\client)
6) Login using your reset password from before
7) A code will be sent to the registered email withy ZOS
8) It may take up to 10-15 minutes for the email with the code to arrive, enter it and NOW YOU CAN LOGIN!
Hope it helps! See ya in Tamriel!