The amphiphilic copolymers of poly(ethylene glycol) methyl ether methacrylate (MPEGMA) and alkyne functionalized 2-hydroxyethyl methacrylate (AlHEMA) were synthesized by controlled atom transfer radical polymerization (ATRP). The reactions were carried out using the standard ATRP initiator ethyl α-bromoisobutyrate, (EiBBr) and the “bio”initiator bromoester derivative of 4-n-butylresorcinol (4nBREBr2). Two substances with antioxidant activity used in cosmetology, (±)-α-lipoic acid (LA) and ferulic acid (FA), were subjected to esterification reactions to introduce azide groups. The “click” reactions between the alkyne contained copolymers and functionalized acids (LA-N3, FA-N3) were performed to obtain polymer-antioxidant conjugates (P((HEMA-click-FA)-co-MPEGMA) and P((HEMA-click-LA)-co-MPEGMA)). The conjugation was performed with an efficiency of 20–75%. In vitro experiments in a phosphate buffer saline (PBS) solution at neutral conditions demonstrated that the sufficient release was reached after 2.5 h for FA and 1 h for LA. The rapid release kinetics as well as the polymer carriers, which were applied to regulate the delivery of antioxidant substances, are beneficial in cosmetology.