Thanks a lot! That did the trick.
For anyone else with the same problem:
$url = 'https://www.website.com/{username}.json';
$data = ['name'=>'George'];
$api_key = CUSTOM_DISCOURSE_API;
$headers = array("Content-Type: multipart/form-data;","Api-Key: $api_key","Api-Username: system",);
$ch = curl_init($url);
curl_setopt($ch, CURLOPT_RETURNTRANSFER, true);
curl_setopt($ch, CURLOPT_CUSTOMREQUEST, "PUT");
curl_setopt($ch, CURLOPT_HTTPHEADER, $headers );
curl_setopt($ch, CURLOPT_POSTFIELDS,http_build_query($data));
$result = curl_exec( $ch );
if ( curl_errno( $ch ) !== 0 ) {
// Handle error, call curl_close( $ch ) and return.
}
curl_close( $ch );
$discourse_user = json_decode( $result );