I don't believe your 21-year-old has a fear of needles, because getting a tattoo would be a nightmare. Your 21-year-old may have a different fear, however, such as a fear of blood or a fear of veins, fear of internal "stuffs" coming out; there are several options that don't even have named because they are lumped in with "fear of needles" but don't necessarily reflect on the needle itself.
I was unable to get through my phobia (I have a fear of veins that eventually spread to a fear of all needles) without help from a therapist who specialized in anxiety disorders. Over the course of several months, this therapist helped me with visualization and relaxation techniques and eventually went with me to get blood drawn (I was pregnant - the reason for my therapy, actually). I'm still probably not someone who could donate blood, but I am able to deal with blood draws and IVs when I need to, especially when it is for the health of my child, and it is because I know how to use these techniques to keep my anxiety at bay.
What was not helpful was people telling me I was being "silly" or to just "get over it." The terror was too deeply ingrained, and so those words were humiliating without being helpful.
If you feel your children truly have a fear of needles or something similar and you don't want to see a therapist, at least read a couple of books about phobias to see whether your kids might fit the bill and learn some gentle exposure techniques yourself. Once you stop belittling your elder children's fears, they may be more willing to put on a brave face for your daughter to help her cope with her concerns.