Any recommendations for a payroll service I can use to pay my Mom's caregiver?

I have no personal experience, but a quick search showed a Reddit thread on payroll services for household employers that you might find useful. Their original question was focused on nannies but the approach should be similar.

https://www.reddit.com/r/NannyEmployers/comments/14kgsd9/payroll_service/

Confirm that they are familiar with your state.

https://secure.aadmm.com/find-a-dmm/

Do a local search for payroll company.