I am (careful now- don't booh or throw stones!) a recruiter and a Navy Dad, so with certainty I can tell you the following because I do this for all my recruits' families:
Your child's recruiter will have the mailing information available to them within 2-4 days after your sailor ships to boot camp. He (or she) should contact you (if they have your phone number given to them by your sailor or yourselves) and give you the information and how to address the envelope. This information is only available to your sailor's recruiter at that particular office, since it is coupled with their SS number. I hope this helps many of you and feel free to pass it on to others as necessary.