Since the care is being provided in the home of your MIL, where you also live, she is not a household employee so would not be issued a W-2.
She would be a self-employed person providing a service. Income she receives for the care of your child would be reported by her on a Schedule C.
You can provide her a Form 1099-NEC or not. She reports the cash income on her return and you report the child care expenses you paid for on your tax return, provided you and your spouse are both working.
