Is it normal for my child to be scratching herself with fingernails?

Yes: Yes, infants will often scratch themselves, sometimes even causing bleeding. It's hard, but try to keep those little nails trimmed. For toddlers and older children, maybe not. If their scratching a lot, especially if it keeps them awake at night, you should talk with your pediatrician about it.

Yes: Babies do not yet have great control over the movements of their arms and legs. They don't mean to scratch themselves, but because they can't always control where their arms go and when they open and close their fists it is easy for those little fingernails to make scratches on their face. Try to keep the nails trimmed (easiest to do after a bath when they are nice and soft) or use mittens.