Is it possible to open a site in the “show webpage” function (or similar), and then return data like a button press from that page?
One approach is to do that sort of thing in/via Scriptable, assuming you construct the page as you can set and evaluate content.
The editor here returns a whole chunk of edited content for example, but the principle is the same.
Not really with Show Webpage. I did a proof of concept on an idea where buttons on a html page generated by a Shortcut call, and pass data back into the same Shortcut using x-callbacks, but a limitation is that the page needs to be URI and opened in Safari, rather than Show WebPage.
Also you mention a site, rather than a page, this might not be what you meant at all.
(Ive not updated this for 13 so not sure if it still works)