Unfortunately, the coding that the site uses to post your questions and return data back to you does not function in that matter. If you want to interact with the Large Language model without using you web browser, you would need to so by using other software or applications to hit their API (you do have to pay for this service) and then return the data back to you. So it is possible, but not in the manner that you are suggesting.
On a side note, I signed up for a 3 month trial of access to hit their API using a session code and a post shortcut on my iPhone. I created a iOS shortcut that allowed me to use OpenAI as a replacement for Siri. It was a fun little project.