is a report of birth an accepted proof of U.S. citizenship?
Yes, We can accept only certain documents as proof of U.S. citizenship and this includes a Certification of Report of Birth DS-1350
I heard I needed a picture ID for them like a state ID or a driver's license but what if I don't have one for them?
In that case we will ask to see other documents. Any documents submitted , must be current not expired and show your name , identifying information,  date of birth or age and preferably a recent photograph. Documents such as an Employee identification card ; School identification card ; Health insurance card not a Medicare card ;  or U.S. military identification card would be acceptable.