It actually depends on what you mean by “daycare.” If you mean a licensed daycare center, then yes, the children will be required to have their vaccinations. If you mean a licensed home daycare, then yes, the children will be required to have vaccinations.
If you mean a local babysitter who is not licensed, then you have no way of knowing because the parents won’t have to submit paperwork and keep it updated, the sitter won’t care because there’s no one from the state coming in to check her paperwork, and if some of the kids are behind on their shots the sitter will have no way of knowing it.
Also, just because the other kids at the daycare are current on their vaccines doesn’t mean your baby won’t get sick. I worked in daycare for several years, and no matter how careful we were about washing hands and sanitizing toys and surfaces, kids got sick. Mostly the illnesses consisted of runny noses, coughs, and ear infections. Sometimes a stomach bug would go through the center, or some other kind of common childhood virus. When you have groups of young children together, no matter how clean the place is or how hard the teachers work to stop the spread of germs, kids will get sick. It is one of the drawbacks of group care. The good news is that by the time the kids go to kindergarten, their immune systems are champs and they rarely get sick. My two oldest kids were in daycare as infants and they’ve hardly had any sick days from elementary school.
You will have sick days because your child will get sick. There’s no way around it. If you are worried that his little system won’t be able to handle it, talk to your pediatrician. You might want to try to find something other than group care until your baby is a little older if he isn’t ready to fight off typical childhood germs.