From some other threads related to this issue:
After testing everything I could possibly think to look at, I did some research - it appears that the "API Key can not be blank" error comes directly from the MailChimp API server and has nothing to do with the code in the plugin. It seems to be a common issue with MailChimp extensions for WP, Joomla, and more. Unfortunately, that also means that I have no way of fixing it.
Caching maybe but I think it may be related to your session. The module stores values in a php session variable. So if your browser page is left open too long or you clear your browser cache and try to submit without refreshing the page this could cause this problem.
If you load the page and try to submit right away does it work?