Can you let me know what page you're testing this from? I just tried from businessaviationnetwork and was immediately logged in when trying the Login with LinkedIn button.
If you're still having issues, caching could definitely be part of the problem. I'd recommend:
* Disabling any caching in the Joomla -> Global Configuration area
* Disable the System - Cache plugin, if enabled
* Disable any other caching or optimization extensions
* Try from a different computer and/or browser
If that doesn't help, let me know what page I can test with and we'll gladly help however we can.
Thanks
Alex