public class Auth
{
  public string Key { set; get; }
  public string pID { set; get; }
  public string DocOrderInts { set; get; }
}

[HttpPost]
public async Task<IHttpActionResult> PostAPI10X(Auth auth)
{
   //Данные надо принять в класс "auth"
}

<?php

$url = 'https://MySite.com/API/';
$data = array('Key' => "XXX", 'pID' => 'YYY', 'DocOrderInts' => '[{"product_id":"10017","vendor_product_id":"23539","vendor_uid":"64374vn"}]');

// use key 'http' even if you send the request to https://...
$options = array(
    'http' => array(
        'header'  => "Content-type: application/x-www-form-urlencoded\r\n",
        'method'  => 'POST',
        'content' => http_build_query($data)
    )
);
$context  = stream_context_create($options);
$result = file_get_contents($url, false, $context);
if ($result === FALSE) { /* Handle error */ }

var_dump($result);