Can UDK4 communicate with a remote server database such as mySQL using PHP?
You have access to the C++ source code, so you can access mySQL with C++. Or you can always write some PHP proxy and call that, if you are so inclined.
I believe what you mean is, can you communicate with a web server, and the answer is absolutely. There are a number of libraries you can build off of, but UE4 has a TCPLink class available for you to be to do exactly that type of communication. I am the author of the HTTPLink from UDK and can vouch for its ability to handle such messaging and processing. If by chance this class is missing it will surely be available as a module to be linked into your project later.