M.K.
We cloth diaper and wipe as well. We make wipes a batch at a time with a little bit of Jason's Homebest baby soap, a few drops of tea tree oil, and water (we keep them in a wipe-warmer). This mix seems to keep bacteria out. It might work to have a wipe container for the wipes and some sort of easy disposal bin for them at daycare (a little garbage can or something?). That way, the wipes are easy for daycare folks to access, and you can take the containers home, wash, and replenish as needed. I totally agree--keep those chemicals off my baby!
If they won't do cloth, it might work to use paper towels that you make this way. That way you know only mild soap and tea tree oil are in them, and daycare folks can just throw them away!