If you want to use Facebook Messenger module - it is required to "whitelist" your shop's domain in "messaging" settings on your fanpage. Below you can find detailed steps where I explain how to do it.

1. open your fanpage on facebook and go to "settings" page, 
2. please open your fanpage on facebook
3. go to settings tab:
   
4. then open "advanced messaging" section
   
5. in field "whitelisted domains" type url to your shop (url of shop where you want to run messenger chat window)