The simple answer to this is to use JavaScript, which is the de facto scripting nature of the web.
Examine the source code of the page in question to find the name of the web form the button is associated with, or maybe even the button itself.
If the button is the submit button (looking at the source code will tell you), then you can emulate submitting the form:
tell application "Safari"
do JavaScript "document.forms[\"myform\"].submit();" in window 1
end tell
(the above assumes the form has a name/ID of 'myform').
If it's just a button not associated with a form, you can click it via:
tell application "Safari"
do JavaScript "document.getElementById(\"myButtonId\").click();" in window 1
end tell
Where 'myButtonId' is the name of the button in question (not necessarily its label)