First - make sure you have the latest patch for Dawn of War.
That fixed some of these spooge error messages.
Second , install the latest drivers for your card - of its ATI - got to
www.ati.com, if its Nvidia -
www.nvidia.com
Now try to start the game - if you get the error again then unfortunately, yes, you probably do need a new card.
This error is frequently given out when trying to run Dawn of War without a videocard capable of performing T&L (Transform and lightning), there's no away around it except buying a new card.
If you decide to upgrade, just post how much money are you willing to spend, and I'm sure there will be some good recommendations made.
But before rushing out to buy a new card, try the steps above, they might make the game work!