Either form is acceptable. It is capturing the same essential information and is not submitted to the IRS, @Opus 17 @SweetieJean and @Carl are correct.
The General instructions of Form W-10, first column state: You may use Form W-10 or any of the other sources listed under Due diligence below to get this information from each provider.
Form W-10 Dependent Care Provider’s Identification and Certification can be used instead of Form W-9 Request for Taxpayer Identification Number and Certification since Form W-10 is a specific form for care providers.
It does not require them to identify their business entity type but otherwise accomplishes the goal: getting the provider's taxpayer identification number, which is needed to file for the Child and Dependent Care Credit.
@richard899
