I think it can be done with keypress/keyup/keydown event with jquery ajax. Just simply send an ajax request to your PHP script which will save or update data to your database on every keypress/keyup/keydown with the value you just typed.
$('#yourTextBoxId').keypress(function(e){
var myValue = $(this).val();
$.ajax({
url:'yourphpfile.php',
type: 'POST',
data:{myValue:myValue}
});
});