SubZeroIRC offers a way to connect to the network without downloading anything!
CGIIRC will run from your webbrower and is basic, but requires no downloads and no other outgoing connections than the one on the webserver.
This is a Java based client, and requires you to have a Java runtime engine installled. It makes a connection on port 6667 from your computer to the IRC server, so you may have to configure your firewall to allow the connection.