That might be relevant if we were talking about personal information. We're not.
"He or she shall obtain the necessary personal information from the best-qualified person and the medical certification from the person responsible for it."
The person responsible for the medical information is the doctor/medical examiner/coroner, and they, not the embalmer, complete the death certificate.
Which doesn't mention vaccination status.
But they talk to them.
It might surprise you but funeral directors/undertakers/morticians/embalmers/mortuary science licensees/mortuary science practitioners talk to their clients about a lot of things that don’t go on the death certificate.
And, as was shown in my previous post, it’s not just a part of being the provider of a service but a part of their official duties to the state to talk to them.