You're right

There is a bug in a code. LinkedIn changed how their behavior and we shouldn't be including a blank API key if none is set. It's a minor code change to fix, if you're up for trying.
In the /components/com_jfbconnect/libraries/provider/linkedin.php file, around line 122, you'll see:
'<script type="text/javascript">' .
$initJs .
"</script>";Simply remove that middle line so it looks like:
'<script type="text/javascript">' .
"</script>";Setting the Javascript API domain should have fixed things. That setting is necessary for some of the other social widgets from LinkedIn to function at all. The Share button can work without an API key though.. so, if you aren't using social login with LinkedIn and don't need to use the other LinkedIn widgets, I'd just remove the key and make that code change.
We'll make an update in our next release to properly detect if no API key is set.
I hope that helps, but if not, just let us know and we'll get you going.
Thanks,
Alex