Just point your REST call to this file and it will store it to a file. Just change $file to whatever file you would like, currently set to request.json. Also remember to chmod that file so it's writable.

REST_endpoint.php

<?php

header("Access-Control-Allow-Orgin: *");
header("Access-Control-Allow-Methods: *");
header("Content-Type: application/json");


$requestData = '';

if(isset($HTTP_RAW_POST_DATA) && strlen($HTTP_RAW_POST_DATA) > 0){
  // POST
  $requestData = file_get_contents('php://input');
  if(strlen($requestData) < 0){
    $requestData = $HTTP_RAW_POST_DATA;
  }
} elseif ($_SERVER['REQUEST_METHOD'] === 'GET') {
  // GET
  $requestData = $_REQUEST;
} elseif ($_SERVER['REQUEST_METHOD'] === 'PUT') {
  // PUT
  $requestData = file_get_contents("php://input");
} elseif ($_SERVER['REQUEST_METHOD'] === 'DELETE'){
  $requestData = 'DELETE CALLED';
}

$file = './request.json';
file_put_contents($file, $requestData);