One way to find a competition is to check the American Homebrewers Association website. http://www.homebrewersassociation.or...ition/calendar
Most of them accept beer, cider, and mead using the BJCP guidelines http://www.bjcp.org/2008styles/catdex.php
You should read the rules for the comp you want to enter. Most of them are similar, but there are differences. For instance, some want all entries in 12 oz brown beer bottles, while others will accept cider in beer, wine, or champagne bottles.
If you need to ship your cider, DO NOT use the post office. It's illegal to ship alcohol with USPS. I usually use Fed Ex or UPS. It's legal, but I don't tell them it's alcohol to avoid any hassles.