Make sure that Dawn of War is closed before trying this. This applet listens on port 6112 on your machine. When you click the button, it sends a packet to a process that's also listening on port 6112 on my server. The web server sends a packet back to this applet. In theory, this verifies that your port 6112 is accessible to the outside world. Tests udp ports 6112, 6500, 27900, 27901, 29910 and tcp ports 6667, 28910, 29900, 29901, 29920.

If it returns "NAT detected" you'll probably be able to play multiplayer games as long as nobody else who has the same issue is in the same game. This means that a router/bridge/firewall between you and the machine hosting this test altered the outgoing UDP packet's source port to something other than 6112.

From the readme

Warhammer 40,000: Dawn of War uses the GameSpy NAT negotiation system to enable most people to play online without configuring their routers, but you may need to manually open ports to play.

GameSpy can require the following ports to be opened:

        UDP   6500
        TCP   6667
        UDP   27900
        UDP   27901
        TCP   28910
        TCP   29900
        TCP   29901
        UDP   29910
        TCP   29920

Warhammer 40,000: Dawn of War uses the following port to host games and accept connections:

        UDP   6112

Check your router documentation for instructions on how to open ports. The process is very different for different brands and even for different models from the same maker.

You could also try putting your computer in your router's DMZ. Again, check your router documentation for details on how to do this.