Hello Chris Chase,
I am really sorry for the delay of this reply as it is a weekend for us here.
I have tried to click "Sign in with Linkedin" from Linkedin autoposting configuration page, then hitting this :
403 Forbidden
You don't have permission to access /administrator/index.php on this server.
Additionally, a 404 Not Found error was encountered while trying to use an ErrorDocument to handle the request.
Hmm, could you mind consult this with your Webhosting provider and see if could help? It seems like when you click "Access" when pop-up the LInkedin account authorization, then it can't redirect back to the configuration page, the token access also can't able to store in database, so it failed.
Keep us updated if Webhosting provider say not related this permission issues.