Sure. You don't need a public internet connection to run a Web app. All you need is a machine with a Web Server and an IP address that is reachable from inside of the network.
If you have a domain name that domain name is generally registered through public DNS which routes through the public internet. If you want only a local domain you can actually do this with internal DNS routing (running an internal DNS server) or by simply modifying the local HOSTS file on the each of hte machines that needs to access that server.
Generally though its more common to use IP Addresses rather than domain names for internal applications.
+++ Rick ---
from Maui, Hawaii