I see this phrase a lot. For me, it roughly equates to > 98%. However, for some I could imagine it meaning > 99.9% or perhaps much lower like > 85%. Or, even worse, is context dependent (the vast majority of pop bottles in the pop machine did not contain pure hydrochloric acid).
I kind of feel it is a cop out on the part of the person using the phrase as is conveys a quantitative value with actually specifying it. It is a lot harder to commit to writing down a concrete number as then the reader expects a reference or more information to back it up. "vast majority" just kind of slips into the brain unexamined.