Can I claim my granddaughter even though the hospital messed up and didn't file the paperwork right so she has no social security card?

[AlanT]

Unfortunately, you will need to wait until the social security card has come since you will need the social security number in order to claim your child as a dependent. When the card comes, you will enter them as a dependent under the dependent section in the program. You will be asked how many months your child has lived with you, please select 'the whole year'. The IRS considers babies born anytime in the tax year as living with you the whole year. 

You can request an extension to file your taxes until the Social Security card arrives.