If you paid for care, you must provide the care provider's tax ID number, that is either her personal social security number or a business tax ID number (anyone can get one if they don't want to reveal their personal SSN.)
Whether or not the care provider then reports and pays income tax is between the and the IRS, it's not your business. But usually, if you report care expense, the IRS will check their return to make sure they report matching income. That's why you need a SSN.
You can claim the credit without an SSN but you can't e-file, and you must attach a written statement identifying the provider and explaining that they refused to give you their tax number, which will pretty much guarantee the provider gets audited.
