Hmmm well it could and it would definitely be an interesting competition, but then comes the problem of if it's your own design or not. It's really easy to just follow directions via a book, and then if you do something more simple in nature, like modular origami, then there's also the possibility that someone else came up with the design separately from you and you didn't know.
hmmmm what about a more general competition, like a Sculpture one - basically you can make any form of sculpture, including origami. That would also increase the likeliness of people to enter as Origami isn't exactly a very common skill.
You could also do something like an "Origami Scene" sort of thing - there the challenge wouldn't be to create a single design, but to organize it all in an aesthetic way - that would also allow people to use designs from books if they have to, but wouldn't give them an advantage.