If you mean
originalfic (ie. non-DB), there are places like fictionpress.com to cater for them. (You could even advertise your original work in the Advertising forum!)
If you mean a place for
better DB fics, the Archive is already inside the Fanfic section to house completed and readable fanfics, and you can nominate the best of them in the recs thread.
If you want the Admin to set up a story contest, you'll have to ask her. She ran a
birthday competition previously, though there weren't many entires. If you want to set up your own competition, there are several threads in the fanfic forum already which aren't story threads, so I daresay you could go for it yourself.
Does this answer your question?