According to supporters of origami, the answer is yes. We tend to agree. The U.S. grants design patents to origami works, and considering the amazing creative work that is expressed in much origami, we believe that such works merit copyright protection. We also believe that instructions for creating such works may be copyrightable depending on the extent of original expression. And finally, compilations of origami works may be protected to the extent of the author's choice and selection of material. Traditional origami is most likely in the public domain.
Getting permission. If you are using protected works as the basis of your cards, you should obtain permission. In 2008, the First International Origami Copyright Meeting (Tokyo) proposed the creation of a clearinghouse for origami artists so that people like you could obtain permission from artists. You might want to contact the participants at that meeting to see if such a clearinghouse was created. If that's not possible, you may wish to check out artist Robert Lang's origami resources page. As for your question about proceeding without permission, our standard answer would apply: if the works are protected and the owner learns about your use, you may have to face the consequences of a cease and desist letter (or worse, a lawsuit). The odds of that happening are affected by the popularity of your work and the tenacity with which the owner pursues infringers.