The first ever AFB vaccine was just created recently. It got US approval only last year (first market to approve it I believe). So yeah, a vaccine exists, but it's not widespread as it's extremely new.

They threw them all in one sentence and that made me think vaccination of bees is forbidden in NZ!

"Burning the hives was more painful when other countries used tools like vaccines, antibiotics and sterilisation - measures prohibited in New Zealand and in some export markets."

vaccine availability can be affected by things like, no one sells the vaccine here, or an application for a vaccines approval being stuck in a pile of papers.