The reason for annual flu vaccinations isn’t because we can’t make vaccines, it’s because the flu virus mutates quickly and the mutations are sufficient to make all existing “knowledge” in the immune system not see it.
The average annual flu vaccine actually includes markers for two to three different strains that are /estimated/ to be the ones that are likely to take off each year. The reason for this years flu being so bad is that they got it wrong.
As far as general immunity, that’s unlikely: immunity basically isn’t inherited - “inherited” immunity is generally derived from the mother’s immune system through milk and some of the microbiome picked up by the baby during birth.
Generally gene editing is unlikely to allow magic vaccines to be built in to your dna.
These days the primary problem in creating vaccines is first finding a set of markers that are sufficiently stable to make it unlikely that they can be trivially mutated away, that also doesn’t match any markers people have on their own cells (otherwise you’d just create an injected autoimmune disorder), and also can be packaged so that trains the immune system correctly.