Yes, the http://mywebsite.com/cgi/formmail reference is the resource that all the form data (the survery responses) will be sent to when the user clicks the Submit button.
The cgi reference in that address stands for "common gateway interface", which was a common way to handle form processing in the earlier days of the Internet. CGI can still be used, but has largely been replaced by newer server-side languages (PHP, Ruby, Python, ...) which are typically faster to execute and have smaller memory requirements. It can really make a huge difference on busy hosting servers.
First/easy thing I would recommend trying: change http://mywebsite.com/cgi/formmail to https://mywebsite.com/cgi/formmail
(Note the "s" added to http)
http was previously the standard for websites, but now https is the standard.
If you host blocked http (port 80) then trying to access the script wouldn't work. Changing to https may be the only change you need.
Does this mean that my server was previously allowing its software to send the form result to the e-mail mentioned in the code,
but that they decided to terminate the service?
Or was another software company involved with the form?
I'm just trying to figure out the best way to proceed.
Assuming changing the FORM action to use an https:// address doesn't solve the problem, you may need to contact your host.
They likely discontinued support for cgi scripts. They may be able to provide you with an alternative address that offers the same support (using a script that they now prefer)
Alternatively, you could make your own script to handle the processing. You'd upload the (e.g. PHP) script to your host and then reference your new script in the html FORM action.
Here's a simple example, but you should be able to find all sorts of examples on Google:
https://blog.getform.io/how-to-create-an-html-form-that-sends-you-an-email/
(There is a section that talks about emailing via the script)