The PHP API relies on the XML web publishing engine. Tick the box to also allow access via XML.
Thanks for the response, Mike. Turns out the server is just fine (as I expected). Once I enabled error reporting on my test server for PHP I noticed I was getting a whole bunch of notices and warnings about deprecated features in PHP within the FileMaker SDK. That's when I realized this project was still using the old SDK form FMS 10 or 11...don't remember which. Anyway, I updated the PHP stuff with the new FM PHP API and now it's all working.