Since you have stated that you're a newbie to Linux, I feel that it's worth pointing out that it's much much less common to need to reboot a Linux box compared to a Windows one.
You shouldn't need to reboot even after updating core software packages. Even if something crashes badly, you can ususally recover without a reboot.
You haven't stated why you'd want to be doing a reboot, but rebooting the whole box really should be an absolute last resort. In fact, rebooting simply to clear an issue is consdered very bad practice for a Linux administrator because it tends to wipe out evidence of what caused the problem, and does nothing to prevent the problem from recurring.
On Linux, most issues that would require a Windows box to be rebooted only require the individual program or service to be restarted.
Finally, a note on security: Doing major system operations such as this via a PHP program is bad security practice because it exposes root level functionality to non-root users. I assume (well, I hope!) that you're planning to lock down access to this PHP page, but even the best secured web page should not be considered secure enough to be running root-level operations.
In short, my advice is that you shouldn't do this. If you must do it, @qbert220's answer should work, but please don't do it.
[EDIT]
With specific regard to changing the IP address from DHCP to static, this should not require a server reboot in Linux. You simply need to restart the networking interface.
Once you've changed the config, something like this should be enough to restart your network interface with the new IP address in place:
sudo /etc/init.d/networking restart
You haven't specified what variety of Linux you're using, but here's a link to a page which details how to do it from the commandline in Ubuntu.
It does require root priviledges though, so you would need to use sudo to achieve it and to add your web user to the sudoers list, which as I said before is really not great from a security point of view.