I have never given my chicks shots, and have never had a problem. But I don't have any other flocks near me from which my chickens could pick up diseases. I also lick the beaters when I make cake or cookies and have never gotten salmonella from my raw, farm-raised eggs. I'm of the opinion (and it's just that - an opinion because I have not researched it) that salmonella tends to happen more often in larger, commercial flocks because they're not as sanitary as most smaller, backyard or farm flocks. Again, this is my opinion only.
What counts as making a chick organic or not depends on both personal definition and, for certified organics, local law as I thought? Not sure!
The vaccines most people talk about your chicks getting is a precautionary shot against a disease that would be guaranteed lethal should your chicken be exposed, so immunization at the earliest possible opportunity is best. If you order chicks from a hatchery, they often offer this done for you for about a buck a pop. It's a bigger concern for large flocks, though! If you only have a small flock and aren't circulating through chickens at a rapid pace, you're not at a big risk of introducing it to your flock. Next time you get chicks, if you're worried, just have 'em immunized. Herd immunization is a thing, even if your original chickens weren't immunized, the majority of their new brethren having gotten the shot will still protect them!
As for Salmonella, that is actually a common name for a very, very varied group of bacteria! Salmonella is everywhere, always. It is actually naturally present in your chicken's gut (and in your own!) and if it wasn't there, you'd miss it. Trust me. The reason chickens are a big risk for spreading salmonella is because they come in contact with their own fecal matter. It's in their bedding, so they are exposed to particles, the moisture, walking through it, getting it everywhere. Salmonella illness is usually just too much salmonella in the system, rather than the presence at all. Feeding your chicken properly keeps their gut fauna, salmonella included, in proper numbers! If you're really worried about maintaining, use probiotics too. Then the rest is all a hygiene element when it comes to preventing you catching sick off it all. Wash your hands and clothes thoroughly, keep the coop clean, and don't lick any chickens