How to allow users to submit the form and pop up the same form after the submission?

then have that same form pop up for a new person (lead) to fill, and repeat the process?

I think you will achieve this with a custom thank you page, for example, instead of showing a thank you message after a submission, you can simply load the form again. 

You only need to provide the link where your form is located on and that should be all, here is also a guide that will help you with detailed info about this: Redirecting-Users-to-a-Different-Page

If you have any question, please let us know.