What is the process for my sister, who will need a caregiver soon, to request her nephew to be her caregiver?

Medicare provides in home services for 60 days following a hospital stay.
These services are usually through the VNA. Did VNA say they would hire him?

No, it's not Medicaid. It's Medicare through which she is going to be provided care.

If you mean she is going to qualify for Medicaid, you need to know your state's rules for Medicaid services. If nephew is getting certified as a CNA or PCA then he might be hired by the agency contracted to provide services.