Is there a nonprescription peridex (chlorhexidine gluconate)?

No. The active ingredient in Peridex (chlorhexidine gluconate) (chlorhexidine) is not available without a prescription. That's primarily because this medicine should not be taken long-term, & it's indicated only for specific oral conditions. If people could just buy it, they would use it indiscriminately & for far too long, so it has to be regulated. There are potential side-effects from improper use which have to be avoided.