I have to say I feel some sympathy for your wife, I was moderately needle phobic, and have passed out having blood tests, I also tensed up once to the point I snapped the needle in my arm. I got into tattoos as I like the final effect, also piercings, I have passed out doing then to myself, but that whole thing passed when I had heart trouble and had a lot of blood tests, I got used to it and the fact I had to live with it, it was that or die and Life became more important, I am now insulin dependant T2 but have never had an issue since, I think it was realising the choice was fear of something to help me live or fear of death the latter became more important.
I think you can ask anybody and they have the right to refuse, at work there can also be a problem with liability insurance for employers, but I think that is irrelevant and common sense should rule