In my situation, the sitter comes to my house.
That is a key detail missing from your first post. Added to a claim that someone lacks a tax ID number...when that person actually DOES have a tax ID number (a social security number is, indeed, a tax ID number)...and it makes me wonder whether I should actually answer any further inquiries, for fear of giving you incorrect information based on incomplete/inaccurate data. GIGO, just like the IRS Withholding Calculator, my friend.
That said, a legal resident who is authorized to work in the USA and possesses an SSN needs to be treated in a manner consistent with the nature of the employment. Treat a person working in your home under your direction as a household employee, and a person working in their own home under their own direction as an independent contractor. The person working (for you or for him/herself) does not need an Employer ID Number (EIN). If you file a Schedule H, you do need to get an EIN.
IN EITHER CASE, the payments to the child care provider can be claimed for the dependent care credit on Form 2441. If the care provider gives you her SSN, full name, address and phone number, you will be able to complete the 2441 in full and file electronically (barring no other issues that would prevent electronic filing). If the care provider refuses to give you her SSN, you will need to keep the W-10 on file and fill out as much of the 2441 as you can.