Each airline has its own policies on this and you haven't stated the age of the child or if this is a domestic flight (your sister is coming from another state inside the US)
If the child is under 2 and traveling as a lap baby, some airlines do require proof of age.
If the child is traveling with a youth or infant priced ticket, some airlines will require proof of age.
Of course to fly internationally you have to have a passport, but from your post, I assume this is a domestic flight.
Outside of the conditions above, there really usually isn't ID required for a child under 18. Best to call the airline to know for sure.
If there is a close relative like a grandparent is living in or near the county where the child was born, they could possibly get the BC at that county's court house for a fee and fax it to her.