Could you not post the URL here so we can go check for ourselves. It seems unlikely that there is no form tag.
Otherwise, no you can't simulate clicking the button with server-side PHP code, you could do so by simulating a browser, but that just get's messy. What I would do is get HTTP request logger (there are a lot of addons for firefox that do so) and then log all your HTTP requests.
And from there you should be able to determine, which URL the data is sent to and how.