Does anyone know is it possible to make clickable button to trigger button on emacs?

I'm newbie. That is I want to do via emacs. Is it possible? I've been dreaming something like the html frame with clickable button (for shell or emacs macro and so on)

please take a look at the widgets package.

You want to create a new button which triggers the action of a different (pre-existing) button? Details?

Could you provide more details on what you are trying to achieve, your question is missing context and I don't really understand what you really want?

If so I assume that it would be a question of finding out which function is getting called when the button is pressed

Load the Hyperbole package. Discover implicit and explicit buttons with arbitrary actions and have your mind blown. If you can handle that sort of thing 😉