Flu shot and blood draw at home?

See if her doctor will order lab work through the VNA. They do blood work and urine cultures for my mom.

Have you contacted the local Visiting Nurse Association?

When my mom was on an at home IV therapy, one of the supplies the company sent was called a Lab in a Box. It had everything needed for the visiting nurse to draw blood if it was needed. Maybe your mom's doctor can order that for her bloodwork?
I would call the doctor's office and ask them for some guidance. I can't imagine your mom is the only patient in this position.
Good luck!