On your live server look closely at the output of phpinfo() and compare it with that from your wamp server.
You may well find that safe_mode is “on”, which will block file_get_contents() from going out, but something else could be blocking external fetches too, so look carefully.
If this is the problem, check if cURL is installed/working on your live server, if so you can use that instead.
After checking Cups recommendation, you may want to check the following:
I haven’t worked with the Groupon API, but a few APIs I have recently implemented required me to tie a domain to the API Key. You may want to check if that is the case with Groupon, if so, you may need a new API Key for your live Server or you may need to update your current API Key.