Ist HPV vererbar?

You can only inherit your own genome, not viruses that are infected with.

At birth transfer… complicated.

So, human papillomaviruses, short HPV, are a huge group of different viruses. They are all related to each other, and they all have in common that they infect the skin and mucous membranes of people.
HP viruses are mostly known from cervical cancer, yes. These are mainly types 16 and 18. It is also vaccinated against it.
But, HP viruses also make genital warts and ‘normal’ warts on hands and feet.

Viruses work in such a way that they target cells and then use them to multiply themselves. You can always only occupy a single cell type. An HP virus, which nests in cornea of hands and feet, does nothing in the mouth.

I’m thinking you’re specifically going out of the HP viruses that have a mother’s mouth, and in the worst case, you can lead to cancer. These viruses can also survive only in similar tissue – mainly in another mother’s mouth, but also in the throat’s head.
Normally, however, both at birth do not come into contact with the mother’s mouth, and also: if the mother is infected, it most likely has antibodies against the viruses, and they are transferred to the child via the placenta and later the mother’s milk, which also has protection.

But, no, if someone is already infected, a vaccination doesn’t bring anything. Because vaccination is nothing but to give someone a controlled, harmless dose of the pathogen so that you have a faster immune response in the future.