Hallo,
the problem is that you are not directly related to your cousin. The WAST will give informations only to the person itself, or, if deceased, to his wife and direct descendants. In your case they ask for a permission (Vollmacht) signed by such a relative in your cousin's direct line. In this permission he or she should explain their descent and agree to your request for informations. WAST expects this agreement within the next four weeks, otherwise they will cancel your request. They also say that once they got this permission, you should know it will take up to 18 months until you get an answer.
Micha