According to BlackHairInformation.com, it is possible, and even common, for black people to get curly perms. However, the process of creating curl in naturally curly hair is quite different than the one used to produce curl in straight Caucasian hair.
To produce curl in straight hair, the hair is first wrapped on perm rods. Perm chemicals are then used to break down and restructure the protein bonds of the hair, resulting in curl. In naturally curly hair, the bonds are first broken through a relaxer application. Then, the hair is wrapped on perm rods and bonded using a special solution to hold the new shape.