If I'm being dense and missing something here, then I hope someone will step in, but...
...if I were a patient having a blood draw, and the phlebotomist asked, "hey, is it okay if this here volunteer sticks you while I supervise?" I would ask about the training the person has had so far. If the answer was, "well, this is his/her training," I would say "no [expletive] way."
If I was having a non-trained volunteer stick me, and I wasn't notified of their lack of training first, I'd sue the hospital.
Mind you, I am a volunteer myself. I don't even help people in and out of wheelchairs, because it's not in my training or my scope of quasi-practice. What local hospital trains people off the street in phlebotomy?