If you want to keep both your children home, find a babysitter that can watch both. If you are happy sending your oldest to a center and are simply worried about him getting sick, I think the best thing you can do is just accept that it will happen. Kids get sick and kids get better. That's how an immune system is built. Yes, he probably will bring some of it home to your family and little one, but that could happen from the shopping cart at the grocery store, playing with friends or family, a kid at the mall, etc. Practice good hand washing, eat healthy foods, and nurse your baby. My second did get more sniffles than my first, but that could be for many reasons. I'm guessing a main one was having an older sister who brought home germs, but it could also have been time of year, different body different immune system, or anything else.
The reasons a daycare center can/not reject an exemption varies by state. If you need help looking into it, post what state you are in.
ETA: Remember that non-vaccinated children get sick and spread germs too. So does your babysitter, your husband, your parents, etc. That's why the best thing you can do is realize that people get sick and that is something our bodies are designed to handle. I only add this because you seem more afraid of vaxxed kids than anyone else, and I'm afraid when your son comes home with some bug (which he almost certainly will) that you will beat yourself up. Getting sick is actually a healthy and normal part of growing up.