If you have a date field named "date signed" that is read only with default value set to "current_date", and that field doesn't exist on any earlier forms in your process, and the signature field is required, you'll basically get the functionality needed to make this work, without having to do any scripting.

That being said - here's a simple script that will populate a date field when the signature field is clicked.  It's far from perfect because it could overwrite the date if the field is clicked at a later date in the process, and it also does not clear the date if the signature is removed.

$(document).ready(function () {
  
  //when the signature field (with CSS Class of signatureDateTest) is clicked
  //the current date is populated into the date field (with CSS Class of dateSignedTest)
  $('.signatureDateTest input').click(function(){
    var myDate = new Date();
    var m = myDate.getMonth()+1;   //month (add 1 because JS defaults months 0-11)
    var d = myDate.getDate();      //day of the month
    var y = myDate.getFullYear();  //4-digit year
    $('.dateSignedTest input').val(m + '/' + d + '/' + y); 
  });  //end of: $('.signatureDateTest input').click(function(){
  
});  //end of:  $(document).ready(function () {