I am sorry, but I think the problem is with the server your website is hosted on. I am getting the following messages from the plugin:
<!-- Request made -->
<!-- An error occured -->
<!-- connect() timed out! -->
<!-- Try manual_cURL() -->
<!-- An error occured: Num 7, Message: couldn't connect to host -->
I have not seen these messages before, but I think it means the server responds too slowly or prevents you from connecting with Instagram. The physical distance of the server from Instagram's servers might be a part of the problem.
Have any other Instagram plugins worked for you?