As far as I know, there isn't a specific area to make up the little pirates. However, throughout many of the stores at the parks and specifically the World of Disney offer pirate costumes and accessories. In fact at the World of Disney there is a room called the Adventure Room where they can Fill-a-Treasure Chest and/or Build-A-Pirate Hat.
Similarly, at the
Bibbidi Bobbidi Boutique located at both Downtown Disney and the Magic Kingdom, boys can be made into Cool Dudes. But as I mom to 3 sons, I think mine would shudder at going into a predominately princess salon. The good news is if this sounds interesting, a similar transformation can be done at the Magic Kingdom where an actual working barber shop resides by the Car Barn on Main Street USA. I know being a cool dude isn't quite as adventurous as being a pirate, it is a chance to have some fun. My sons have done it many times (and my husband, too, when he had hair!) and loved the bright Mickey on the back of their heads.
Finally, are you familiar with the pirate cruises? Little Landlubbers (ages 4 to 12) go on a 2-hour hunt for treasure and adventure on the high seas of Disney's Seven Seas Lagoon and Bay Lake. The boat, with its Jolly Roger raised high on its mast, sets sail for one port of call after another (other Magic Kingdom area Resorts). When the ship runs aground, the pirates get to run amok, searching for buried treasure in designated areas and dividing up the loot they uncover. A hearty lunch, a special pirate bandana and lessons in pirate catch-phrases are provided for the adventure. Reservations are suggested and can be made by calling 407-WDW-DINE.