Theoretically, if your Apache server user (the user Apache executes under) has the correct permissions this should be possible.
You would however need to make a $HOME directory for this user because allot of applications store data within hidden dirs within $HOME.
A better option might be to setup the suexec module for Apache, and have your users own there own versions of this 'site' interface. Then your server could execute commands as the user who owns the sites.
The entire problem is that websites run under the server, while applications need to run under indeveidual users.
I'd like to say this is simple to fix but its likely not unless you know a bit about Linux and Apache.