The point is that cURL is used to send HTTP requests directly from within your executing PHP script to whatever URL you want, which is typically what you want to do when accessing an API. I was just giving an example of how you might dynamically build a URL string (being sure to url-encode things that might have to be encoded) in order to create the URL to be transmitted. Normally you don't want to send the exact same request data every time -- usually there is something dynamic going on -- if not, then just ignore that part.
As to whether or not cURL is enabled in your installation, I have no way to know. I suppose you could just try a simple script that tries to do a curl_init(), and see if it blows up or not. If it is, in fact, not enabled/installed, then the options are to get it installed, find out if the url_fopen option is enabled (usually not due to security issues) and just put the url into a file_get_contents() call, or switch to another host.
Or the other alternative is that I'm totally misunderstanding the situation and what you want to do.