Is there a German web site that has all the Lutheran church records of birth? I don’t know if Anna Maria was Catholic but her husband was not.
Where should I look for her mother?
I haven't got a clue. Frankfurt is not the area I have to do research by myself and I don't know the parish there, but probably some others can help you. At first you have to try to find the death certificate from Anna Maria (maybe her birthdate is on it) and then the baptism/ birth record in Frankfurt/Germany.
All I have found was:
Name: Friederich Christoph Luckhardt
Birth Date: 23 Mrz 1802 (23 Mar 1802)
Baptism Date: 23 Mrz 1802 (23 Mar 1802)
Baptism Place: Evangelisch, Ulm, Donaukreis, Wuerttemberg
Mother: Maria Justina Luckhardt
FHL Film Number: 1184821
Reference ID: 2:27TF3JQ
There again... in this area there were lots of Luckhardts. Not alot of records of that area seem to be transcripted yet.