Specialist is a weird term, in general you use "expert" or something like that in science for someone who is well versed in his field. In this context, probably a person who has studied things like cellular, molecular biology, is an MD or PhD and has published several articles in the field.
But in general, science doesn't rely on authorities, so there is really no single "vaccine expert" that has all the answers. Expertise is something you earn from contributing to advances in a scientific field.