You need to order a copy of your mother naturalization certificate (the moment you became a US Citizen). A DR birth certificate won't do you any good, do you even know your SS number?
He can NOT order a copy of his/her mother birth certificate. For one it cost a few hundred dollars; application needs to be signed her mother; picture of the mother needs to be sent, and she will need to provide important private information about her mother that she may not know.
A Dominican Birth Certificate is very important to get a U.S. passport if he doesn't have a Certificate of Citizenship, it proves that she is the daughter of the U.S. citizenship.
How Does the Child Get a Passport Under the Child Citizenship Act?
You will need the following when the child applies for a passport:
Proof of the child''s relationship to the American citizen parent.
For the biological child of the American citizen this will be a certified copy of the foreign birth certificate (and translation if not in English). For an adopted child, it is a certified copy of the final adoption decree (and translation if not in English);
The child''s foreign passport showing the Bureau of Citizenship and Immigration Services in the Department of Homeland Security (USCIS) I-551 stamp in the passport, or the child''s permanent resident card (green card);
Proof of identity of the American citizen parent(s)
Passport application, passport photographs and fees. Go to UPassport ServicesU for forms and full instructions.