Simple answer: It is NOT mandatory that you issue your day care provider a W-2. It is mandatory that you report her name, SS#, and the amount you paid her on IRS form 2441, which is filed with your tax return when you use tax free FSA benefits (or claim the day care credit).
As the other replies indicate, there are other complications and issuing her a W-2 helps reduce those. The fact that she has no other income, impacts whether she has to file a tax return and whether you may hear from the IRS about your FSA benefits.
If you paid her more than $4200 for 2019, she cannot be your dependent (she fails the income test).
