I hope that your provider does follow your wishes now, but if not, I would remove your son. Not because she feeds him PB or popcorn, but because she doesn't have respect for your wishes.
I know with the USDA food program, peanut butter is allowable, as is popcorn. So I have fed my daycare children both. Since starting to focus on toddler care I don't serve either as with PB it is mess, and popcorn they don't finish a whole bag and there is a lot of waste, but not because either is a risk. If I had a child with a family background as yours, I certainly wouldn't give that child PB, especially after you request not to! My own 19m old son has had both for months. I don't mind cleaning up one messy toddler covered in peanut butter as much as cleaning up 3-4! Haha! That being said, neither I nor my husband have much beyond my own seasonal allergies. If there was a history of allergies, then of course I wouldn't give my son high allergen foods.
It is often a big hassle and pain to have to accomodate every parents request. With the food program (if she is in it), there are steps to go through if your provider refuses to follow your requests. First of all, if there is no true medical reason, you can only request, but since it appears your son may have an allergy, you can have a form signed by your doctor stating that he cannot have peanut butter.
Having said all that, I pride myself on being a very "accomodating" daycare provider, and while I have only had one request for a special diet, and the mom actually provided the foods (they were trying to avoid artificial dyes to see if they caused her daughter's behaviour issues... turned out she had ADHD), I do try to follow any of my parents' wishes from special diets to avoiding cold temps for a previous RSV sufferer to keeping a child awake so they would nap for a long car ride and so on. So in my own opinion, if your provider continues to refuse to follow such simple requests as no PB and no popcorn, I would seek care elsewhere as there will probably be issues with other aspects of your sons care down the road.
Also, if you do have issues and decide to stay at home, I highly recommend child care! I myself have only 4 extra kiddos, and 2 of them are part time and only overlap by a few hours a week. I make enough to pay for groceries and such (diapers, cleaning supplies, etc) as well as my student loan and car loan. Not a huge money making business, but I love it, and I am then able to stay home with my children. There is no rule that says you have to take 12 kids or anything. Just taking on one or two other children around your son's age would work great.