"Our vaccines are working EXCEPTIONALLY WELL" except...but...

EXCEPTIONALLY: To an unusual, remarkable or exceptional degree.
WELL: In a good or proper or satisfactory manner or to a high standard (`good' is a nonstandard dialectal variant for `well')

According to these definitions, I think it's more appropriate to say that the vaccines may work moderately well in the short term, and we have no idea how well in the long term. For all we know, they may work exceptionally bad in the long run, like maybe they'll kill you. That's better information than you will get from the CDC.