is she a United States citizen?

they should both become naturalized that way they wont have this problem anymore and being that their father is American it should be easier..

It most likely depends on her mother. If her mother became a naturalized citizen when she was quite young she may have been included in that paperwork. If her mother came over with a green card she may have become a permanent resident and neither would be citizens. It should be fairly easy to find out whether she is a permanent resident or a citizen. Try calling an attorney that deals with citizenship to see where you would verify? A permanent resident with a green card can have a social security card, drivers license, etc. They are legally allowed to work in the U.S. and must pay taxes, etc.

I had a friend that came to the states at age 6. Parents became citizens after child was 18, so child never became citizen but remained permanent resident with social security card and DL, etc.

If they haven't been naturalized properly, they may not be American citizens. But given the history of their lives, I find it difficult to believe they aren't. Either way, it sounds like they are definitely legal (social security number, driving license, serving in Vietnam, etc). She can check her status and get naturalized.

German Citizen

Sprechen Sie deutsche?

If she never became a citizen in record she is not one. It does not matter how long she has been here.