PHP | Запрос по типу Callback API к php скрипту

Используйте curl. Пример

// Sets our destination URL
$endpoint_url = 'https://somesite.com/path/to/endpoint';
// Creates our data array that we want to post to the endpoint
$data_to_post = [
    'field1' => 'foo',
    'field2' => 'bar',
    'field3' => 'spam',
    'field4' => 'eggs',
];
// Sets our options array so we can assign them all at once
$options = [
    CURLOPT_URL        => $endpoint_url,
    CURLOPT_POST       => true,
    CURLOPT_POSTFIELDS => $data_to_post, //json_encode($data_to_post)
];

// Initiates the cURL object
$curl = curl_init();
// Assigns our options
curl_setopt_array($curl, $options);
// Executes the cURL POST
$results = curl_exec($curl);
// Be kind, tidy up!
curl_close($curl);