May a notary make a certified copy of a birth certificate or a marriage license?

No. Birth certificates and marriage licenses are publically recordable documents. Recordable documents are recorded with some specific governmental entity, such as the secretary of state's office, a court of law, a county clerk, or the Bureau of Vital Statistics. A certified copy of a recordable document may be obtained by contacting the recording entity. A notary cannot make certified copies of recordable documents. A notary may, however, make a certified copy of a non-recordable document. A non-recordable document is one that cannot be recorded with any type of governmental entity. For instance, a letter is not recorded with anyone, but there are times the sender of the letter would like to maintain a certified copy of that letter for his or her file.

Source: Texas Secretary of State