Vasovagal response, unless something traumatic happened in the past that causes the problem, and then it could be a post-traumatic reaction or response (such as someone who had an event in their past where they had to get a lot of shots--not just allergy or diabetic injections, but a trauma like an accident or injury, so that it's tied in with that experience).
Most responses are neurological and can be overcome with good relaxation techniques-you hate to say mind-over-matter as if they are weak-minded, but really if they are able to get past the fear, the body will stop shutting down.
Or simply Shell Shock
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