I totally get where your daughter is coming from on an emotional level. I used to be the same with blood tests (and it's not under your conscious control, I've actually had situations where I've been consciously looking at posters on the wall thinking "la la la, distract yourself, think of something else, it'll all be fine", thought I was perfectly relaxed - by my standards - then, boom! black out time).
But it's not a good enough reason not to have them. It's a phobia (like mine) which needs to be controlled for your own good as a person. Lots of medical procedures necessarily involve blood tests/ needles, you can't escape from that unless you lead a remarkably charmed life.
I'd sit her down and talk through it with her, listening to her, but stressing that needles in the course of health care aren't just going to go away. Then see if you can get it done at the GP's surgery, and explain before hand that she needs a sympathetic HCP, and preferably a chance to lie down flat (will minimise the risk of blacking out). My experience with phlebotomists is they'd much sooner know up front and let you have the extra minute or two it takes to arrange you on the bed rather than in a chair, rather than have to scrape you off the floor later.
And I repeat: these things are not rational. The fainting is a physiological response which isn't under your control, you can't think it away. (With time and patience from HCPs it becomes less likely, and you do start to get a modicum of control.) But you do (generic you) have to learn techniques to mitigate the situation.