Do you mean actually click that button in user's browser or just perform the same action as if user would click it (for example, get search results in your PHP program)?
You definitely cannot do the first as you'll need to use some javascript. And in fact even javascript might not be possible due to security restrictions (you can't access a page loaded from different domain).
As for simulating the browser in PHP - this is very possible as this is esssentially sending a HTTP request. You can either build the request yourself and send from PHP to the server using one of socket PHP extensions, curl or fopen wrappper. Or you can use one of several available PHP libraries such as Guzzle ( http://guzzle.readthedocs.org/en/latest/ ) or Symfony BrowserKit ( http://symfony.com/components/BrowserKit ) .