Yes, you will be able to enter LAFCP for the provider's tax ID. On the screen where you enter the provider information, at the top where it asks for the tax ID number, you can select EIN, Social Security number, or Other. Select Other. Then for the type of provider select Individual and "Foreign provider." TurboTax will insert LAFCP in the space for the provider's tax ID on Form 2441.
When entering the foreign care provider's address, for the state select "Foreign or U.S. Territory" at the very bottom of the drop-down list. Leave the Zip code blank.