php - How to keep form values after submitting to another website?

You can use JavaScript to push needed values to cookies before, as form send. For example:

<form onsubmit="return storeValues(this);" action="" method="POST" name="userForm">
    <input type="text" value="" name="firstname">
    <input type="text" value="" name="lastname">
    <input type="submit" value="Send request">
</form>

Now JavaScript side:

<script>
  /* Set cookies to browser */
  function storeValues(form)  
  {
    setCookie("firstname", form.firstname.value);
    setCookie("lastname", form.lastname.value);
    return true;
  }

  var today = new Date();
  var expiry = new Date(today.getTime() + 30 * 24 * 3600 * 1000); // today + 30 days

  function setCookie(name, value)
  {
    document.cookie=name + "=" + escape(value) + "; path=/; expires=" + expiry.toGMTString();
  }
  /* --Set cookies to browser */

  function getCookie(name) {
    var value = "; " + document.cookie;
    var parts = value.split("; " + name + "=");
    if (parts.length == 2) return parts.pop().split(";").shift();
  }

  /* Loading cookie values into the form */
  if(firstname = getCookie("firstname")) document.userForm.firstname.value = firstname;
  if(lastname = getCookie("lastname")) document.userForm.lastname.value = lastname;
  /* --Loading cookie values into the form */

</script>

You can get more info about setting cookies using this example: http://www.the-art-of-web.com/javascript/setcookie/