ERD50, you already got info about official appeal - that is ONLY way to go, telling as a person involved in claim processing as 3d party. Make sure put keyword 'APPEAL' in the header of your letter, describe situation in facts line by line (bullet points) an reference each EOB, attach copy of your insurance card, copy of the receipt, include all copies of denial EOBs, send by certified letter.
All appeals are registered by insurance company on day when it received, and they have 30 calendar days to review and answer - that is legal requirement and all of them very paranoid to be on time as they get audited for that. All appeals are handled by separate unit that work only with appeals - so they would not know all the story unless you write it clearly in the letter, that will give them enough info to investigate. If it is denied they will give you written instruction how to submit second level of appeal, do not hesitate to submit it - at this point decision will be made not by single person but most likely by the panel. Simple letter from your PCP that you are highly benefited from that vaccinations and would be refereed to do it if asked will go long way also.