You are correct that bronze and iron-age folks did not have access to vaccines. That was also a time in which a majority of children died in infancy, women routinely died in childbirth, and the life expectancy for adults was short. People routinely died of diseases considered marginal now.
There are lots of things - not just medicine - that we have now that did not exist in antiquity. Electricity, mass-produced clothing and automobiles are a start. If you want to live the way biblical figures did, I hope you are wearing homespun clothes, growing your own food and riding donkeys to town. Just be aware that if you're refusing to vaccinate your kids, you're not just endangering their health, you're also endangering the lives of other children they may come into contact with. There are parents of kids with leukemia, for example, who are terrified that their kids will contract the measles and die because of people like you.