Phobias are scary, and no matter how much we tell ourselves that they're irrational and silly, we can't get rid of the fact that they're terrifying. And that's okay. You don't need to "grow up", your behavior is not "unacceptable", and while this is a problem, it's not your fault. You didn't ask to develop this phobia. You didn't wake up and think, "I think I'll have a phobia of needles".
You do need to deal with this, but "sucking it up" is not going to fix anything. Time, therapy, and patience will. Just like a broken bone, phobias can change the way your daily life plays out, and pushing yourself too hard and getting frustrated when your phobias aren't 100% gone is just like trying to run on that broken ankle and getting frustrated when it doesn’t heal.
What you need, first and foremost, is a better support system. Your family and friends don't understand phobias. Plain and simple. They think it's just something childish, something you can get over with "courage and a good attitude". But it doesn't work that way. You need to show them what it means to have a phobia. Sit down and talk with them about what it's really like, and educate them. If that doesn't work, then you need to get out of that situation as soon as you can and find better people who will support you the way you need to be in order to get rid of this phobia.
You do need to get your vaccinations, and you do need to defeat this phobia. But acting like it's not a big deal and shoving yourself into a scary situation is not going to solve a single thing. It's going to make it worse. This is a psychological problem--fix it by getting therapy, not by trying to convince yourself that you're being a child. You aren't. Phobias happen to everyone, regardless of age or situation. This isn't a case of "being brave". This is a case of "I'm struggling and the only way to fix this is through therapy".