You should have gotten her SSN or Tax ID when she began to provide care for your child.   Unless you can enter that information you have a big problem.

When a childcare provider will not give you a SSN or Tax ID

If they still refuse to provide an EIN or SSN, send your provider a W-10 (Request for Taxpayer Information and Identification Number).  You can get a copy of the W-10 by clicking here: W-10 Form & Instructions  Complete as much of the form as you can and send it via certified mail (after making copies for yourself).

If the provider does not respond within 10 business days, you can still file your return.  However, you won't be able to e-file.  Print and mail in your return, along with an explanation and a copy of the W-10 you sent your provider.

https://www.irs.gov/individuals/international-taxpayers/taxpayer-identification-numbers-tin

No, you need to provide a social security number for the babysitter to claim the credit.

"Can I get the credit." 

Maybe.  You should have collected their SSN on a W-10 before you hired them.  If they provide the SSN now, you can claim the credit.  If they refuse, you can file by mail and attach a written statement.  You must explain your diligent efforts to obtain the provider's SSN, and you must give as much identifying information about the provider as you can (name, old and new address, phone number, etc.). This IRS will review your statement on a case by case basis and may award the credit, but it is not guaranteed.  

Turbotax might have a check box for "provider refused" that will allow you to finish and print your return.  (I haven't tested it this year.). If not, use a fake SSN and then cover it up with white-out on the printed copy, write "refused" and mail your return with your statement.