I had the same problem in combination with the plugin “Price Based on Country for WooCommerce”. Downgrading the Amazon Pay plugin to 1.11.1 solved it for now.
Thanks, the linked topic helped: It’s not an ideal solution, but downgrading the Amazon Pay Plugin to version 1.11.1 has solved the problem for now.
Update: It was related to file permissions. Got it back to work by myself.
Solved.
My new hosting provider (I migrated my blog 2 weeks ago) blocked the plugin from executing scripts on the following servers.
http://stats.wordpress.com/
https://dashboard.wordpress.com/
Security reasons, as they said.
After a few emails to customer support everything is working fine again.