No, no need for Form 1099-MISC. If you qualify to claim Child and Dependent Care Credit, you can claim it without issuing 1099-MISC. However, you will need name, address and tax ID (either SS# or EIN) of the provider.
Go to Federal Taxes (Personal in Home & Business version) - Deductions & Credits - in the list of Deductions & Credits find Child and Dependent Care Credit and answer all the questions. Before doing that, make sure that you have entered dependent's information under Personal Info.
You can read more about this credit here: https://ttlc.intuit.com/replies/3301418
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