We don't know where this took place- this could have been a place where there were very few blacks around, never mind a black church or community.
I concede that there may have been an agency, even several, that rebuffed this girl. But even if she had to travel a couple of hundred miles to find another agency, a black church, or a black family, she would have done that if she had wanted to have the baby and do the right thing. Plus, there are good people (black and white) you can turn to even in neighborhoods that seem unfriendly to minorities. For example, are we supposed to believe that if she'd gone to priest at a Catholic church -- or a minister at a Lutheran church, etc, -- she would have been turned away? I don't buy her story.