Yes, it is called AJAX.  : )
In jQuery, for brevity:
// or $('#textbox_autopost').blur if you want to do it when the box loses focus
$('#textbox_autopost').change(function(){
    $.ajax({
        type: "POST",
        url: "some.php",
       data: {text:$(this).val()}
    });
});
if you want to do it via button click
<button id="inlinesubmit_button" type="button">submit</submit>
$('#inlinesubmit_button').click(function(){
    $.ajax({
        type: "POST",
        url: "some.php",
       data: {text:$('#textbox').val()}
    });
});
You can also do it through an A HREF (or submit button, or  or something else wacky:
<!-- backup if JS not available -->
<a href="handler.php" id="inline_submit_a">add comment</a>
$('#inline_submit_a').click(function(evt){
    $.ajax({
        type: "POST",
        url: "some.php",
       data: {text:$('#textbox').val()}
    });
    evt.preventDefault();
    return false;
});
If you want to do it on enter:
$('#textbox_autopost_onenter').keydown(function(evt){
    if ((evt.keyCode) && (evt.keyCode == 13))
    {
      $.ajax({
        type: "POST",
        url: "some.php",
        data: {text:$(this).val()}
      });
      evt.preventDefault();
      return false;
    }
});
Final, site-ready code:   
$(document).ready(function(){
   function submitMe(selector)
   {
        $.ajax({
          type: "POST",
          url: "some.php",
          data: {text:$(selector).val()}
        });
   }
   $('#textbox_button').click(function(){
      submitMe('#textbox');
   });
   $('#textbox').keydown(function(evt){
      if ((evt.keyCode) &&(evt.keyCode == 13))
      {
         submitMe('#textbox');
         evt.preventDefault();
         return false;
      }
   });