As of 4.0.3.3, the API Key is now required.
I've modified the integration to add it as a parameter in the module and put it into the API call.
Module's XML gets this, and then it's used in the prep for the actual call:
<field name="api_key" type="text" label="API Key" description="The Sendy API Key (required for API calls as of 4.0.3.3)" default="" />
I'd be happy to post up that code if asked, but I suspect SourceCoast can add it as well. It took me about 20 minutes. If so, I can do it in a week (I'm about to leave for Amazon AWS:ReInvent in Vegas, so I can package it up and post when I get back). But again, it's a very trivial change, so I suspect SC will do it before I get back