So, in the story I’m writing I have two characters who, due to the circumstances of their births, don’t have birth certificates. Both were born overseas, one in Germany and the other in Romania. The story demands that they immigrate to the United States (around 1985, if that makes a difference.) I know you need your birth certificate for something as official and legal as immigration, and then the younger one (who is only a baby when he comes to the U.S.) would need it later for school, college, getting a driver’s license, etc. What happens in this case? What sort of document takes the place of the birth certificate to fulfill the requirements?
My grandmother was born in the 1920s in rural Idaho and never had a birth certificate. When she travelled out of the US, she used her religious baptism certificate as proof of her birth - it had the date, location, and names of adult witnesses. Funny thing, though, being LDS, she got baptized at age 8, not as an infant, so her passport says she is 8 years younger than she really is.
Infant baptism is very common - when a certificate of birth is not available, the US government recognises Christening as proof of birth.
Oooh, thank you, Ronia. 
These characters are definitely religious and would have been baptized. The certificates would be in Romanian and German, of course, but I’m sure that immigration people are familiar with the idea of foreign languages.
I don’t have a birth certificate. In the States, I use my certificate of naturalization to get my passport, then used my passport to get all of the rest of my ID. The naturalization certificate I got because my [adoptive] parents were already citizens, of course, and they brought me over.
That works for kids, though, but once you can get the certificate, you’re pretty much good to go.