So can I submit the form data to the same CGI Perl script? Or do I need to have one CGI for the HTML Form, and another CGI to process the form and save to the JSON file?

You can submit the form data to the same CGI Perl script. It is quite a common pattern for one script to act as both form generator and form handler for precisely this reason. There are also a whole bunch of tools on CPAN to help you accommodate this. I have used a combination of WWW::Form and WWW::FieldValidator to good effect but there are plenty of others too.