The easiest thing to do is to ask your host to open up the PHP directive allow_url_fopen. That is how PHP is able to issue an fopen (or any sub-variation of it including file_get_contents() and file()) over an http:// protocol instead of a file:// protocol. Baring that, you can use either curl library or directly invoking sockets to connect remotely. Curl can easily be disabled by simply not configuring it, while sockets are the least likely to be closed since they take explicit commanding on the denied functions list.
If they refuse to open the allow_url_fopen directive, than check if your sockets are available:
$rf = new ReflectionFunction("fsockopen");
printf("fsockopen is available? %s" . PHP_EOL, ($rf->isDisabled() ? 'no' : 'yes'));
If that says its available, you should be able to use sockets for this.