@xmasbaby0 's answer applies to claiming the child care credit only.
@AmeliesUncle 's answer is actually more important, because missing the credit means missing the credit, but you can be penalized if you don't issue a W-2 when it is required. If you pay a nanny to work in your home, and you pay them more than $2200 per year, you are required to issue a W-2 and pay household employee tax. You are supposed to get their social security number before they start working for you.
If you can't get the SSN by December 31, there may be a way to issue one without the SSN, but that's complicated and hopefully you will work it out before then.
