I can't speak for a specific place to go, but yes this does happen. I would check with the laws in IL and find out what the ratios are for your area. 8-1 infants sounds pretty high, and I am from a state with some of the worst daycare laws out there.
If they require you to bring diapers and wipes and it HAS to be yours for your child's sensitive skin, you may need to occasionally check the level to ensure that they always have some (maybe keep some in your car incase you notice in the morning). But since you had diapers there, how do they know whose are whose? Mine has cubbies with the kid's names on them, another place had us put the first initial on the diaper, etc. I would talk with the teacher and let her know why it is so important. Also, the director should never brush you off (I have only learned that because I am now at a good daycare).
I use cloth diapers, and have had issues when new teachers come, of them not being tight enough, or changed often enough. Some things, I am okay talking to the teacher about, other things (there were still pieces of poop on his bottom), I talked to the director about. For the last occasion, the teacher was moved to a class with older students, because she was great, but diapering seemed too much.
When you find a new place, I would make sure that under allergies, you list an issue with diapers. They may take it a little more seriously. I would also ask questions as to how they ensure that only her diapers will be used.