PHP is a server side script, which means it runs on the server, and sends its output ot the client's browser.
PHP cannot communicate with anything on the client machine.
PHP can output javascript or another language.
The only thing that might be possible,,,,
Can you connect to your Palm Pilot from another computer and make it sync? If so you could learn the proto, and open a socket to do it.
Even with your example, what is the use? Why do you want a webpage to tell your Plam to sync? I do not think you have got the concept of AvantGo down in your head. Wouldn't it be easier to just cradle it and press the button?
It looks to me like software like AvantGo is using wireless internet, not web -> (sync) -> palmpilot.
( Thats a iPAQ and not a Palm Pilot, there is a HUGE difference there. )