上記のリンクの二つ目に、以下のように書いてあります。
Severe reactions to the Pfizer Covid-19 vaccine being used in New Zealand are rare, but when they do happen, these events - known as ‘treatment injuries’ - will be covered by ACC if the criteria for treatment injury are met.
The ACC Treatment Injury Guide contains three core requirements that need to be met in order for a claim to be accepted.
Firstly, the injury has to result in physical harm or damage to the patient. Next, the injury must have been caused by the treatment - in this case the vaccine - and finally, the injury cannot be an ordinary consequence of getting a vaccination.
The more common, mild reactions that some people experience when they get a vaccine, such as inflammation around the site of the injection, is unlikely to be covered.
But infections such as cellulitis or septic arthritis due to the vaccination, or anaphylaxis, are not ordinary consequences and are likely to be covered.
上記を読む限り絶対ではないけどカバーされると思いますので治療を受けたところに確認してみるのが良いと思いますよ。