I am experiencing the same problem. I have selected the contacts that I would like to export and then I click the 'gear' icon in the lower left of the screen. I then click "Export vCard" and nothing happens - no 'save' windows to set a save location, no pop-up blocker warnings - nothing. It appears as if the apple server is just not responding.
I have tried using the most recent versions of both the Chrome and Internet Explorer browsers and the situation is the same.
I guess it wouldn't be the true Apple experience if things just worked the way they should, but iCloud never seems to work for some reason or other and it is getting very frustrating.
Any suggestions?