Said companies already have your bank account details, TFN, probably passport number, home address, telephone number, next of kin, super details, and your daily routine (because they know when you are at work). If you don't trust your employer it's already way too late. Seriously get over it, they have more than enough info to raid your accounts and steal your ID, knowing your vaccination status is the very least of your worries. Pretty sure airlines now are required to have your health details for flight crew and airside workers for damp, at least any medications you are taking must be disclosed.
Not sure what planet you live on but you surrendered most of your privacy rights just by posting on this board, let alone being linked to a social media platform or even just owning a mobile phone or connected device, you just don't know it.