jQuery function to display an alert when submitting blank fields
Your code can be refactor in multiple ways. To remove code repetition, create a new function for repeated code and use it as a callback of each filter.

function someFunc(e) {
  if (this.value === "") {
    alert('Please Fill In Or Remove The Blank Fields First')
    e.preventDefault();
    should_run = 0
  }
}


Use above function as

$('.key_field:not(:first)').filter(someFunc);


This will automatically remove redundant code as well as reduce number lines.

Suggestions:

Instead of playing around DOM and get value for each field separately, try to use serializeArray method of JQuery. See example

Please make habit to use ; to terminate line.

Hope this is helpful.