In the old days, you could use
mod_access_identd. It added an
Ident directive allowing authentication and authorization through the Ident protocol. The Apache server interrogates an Ident server running on the client machine, which can provide reliable information as long as you trust root on the client machine and the communication between the client and the server.
Nowadays, on the Internet, both assumptions are almost always false: communications can't be trusted without cryptography, and being root on the remote machine is not significant. In your case, Ident would work. I don't see any signs that the module has been ported to Apache 2; I don't know how hard it would be to port it. You would have to run an
identd server; most Linux distributions ship a couple but don't install them by default.
There's still an official
mod_ident, but it only provides logging, not authorization.