For that I don't see why you would need the card, since you have your 2nd tag which you were only able to purchase after being drawn for that hunt. But then again, how would a game warden know which hunt you drew? That won't be shown on the tag. So I take that back, you probably do need the card.
Try calling 360-902-2515. That's the WDFW in Olympia. I bet they can somehow get you a duplicate. I know one year I never got my card in the mail. I went in to the Olympia office and they printed it out for me, just on 8-1/2 x 11 size paper.