This is an often asked question and the only answer I've been able to come up with is Speech Language Pathologist, if we are talking about Patient Care. An argument could be made for MSW and Psychologist, but I don't know if you consider that in the "medical field" and you could still deal with patients getting injections while you are in the room or nearby.
Pretty much all other Patient Care positions will see needles or deal with needles at some point during their career. Yes, this includes Physical Therapist, Occupational Therapist, Pediatrician, Medical Assistant, CNA, RN, and LPN.
Any kind of radiology tech (sonography, ultrasound, MRI, CT, x-ray) will also have to deal with needles at some point. They may have to shoot a film during a surgery, during an emergency, etc. They have to know how to run an IV.
It all comes down to how deep your fear of needles is. If you can see other people use them and give injections to a patient, then your options are more open. If you can't even see a needle without passing out or getting woozy, then your options are severely limited.
Edit: RD may work, but you also have the chance of dealing with patients with diabetes and you have to help them monitor glucose levels in your presence or even to help give insulin. This isn't necessarily something you are trained in in school, but it can become a part of the job by default.