We don't accept volunteer phlebotomists. The only hospital I know of that does is the military hospitals, and you volunteer through the Red Cross located inside the military hospital.
I work full time as a phleb. We have trained phlebotomists on the job. But, we have far more success training people for the job, then training them from scratch. The first one we hired (since I started working there), decided that she didn't like getting up at 3 in the morning to come to work, so she was fired. The next couple aren't terribly good, and we all shudder when the new people get stuck working a shift together without an experienced phleb there.
The latest hire that we trained on the job is more of a burden than a help, really. She comes in at 4, and by the time 6:30 comes around, she's done maybe 10 draws, and missed half of them. Granted, I wasn't perfect when I started either, but since we're expected to do 10 draws an hour (on our slowest days, I usually average around 12 an hour), it's far more irritating when everyone else finishes their draws before she even gets halfway through hers. She's also a generally annoying person, and it doesn't seem like she really wants to work, as once we all come up to pick up her slack, she'll disappear.
So, by all means, apply to be a phlebotomist, and hope that you can get hired. But please actually make an effort, or you'll likely be hated by your peers, especially when they're like us and already short-handed.