In this work, a commercial SBS was functionalized with the 2,2,6,6-tetramethylpiperidin-N-oxyl stable radical (TEMPO) via free-radical activation initiated with benzoyl peroxide (BPO). The obtained macroinitiator was used to graft both vinylbenzyl chloride (VBC) and styrene/VBC random copolymer chains from SBS to create g-VBC-x and g-VBC-x-co-Sty-z graft copolymers, respectively. The controlled nature of the polymerization as well as the use of a solvent allowed us to reduce the extent of the formation of the unwanted, non-grafted (co)polymer, thereby facilitating the graft copolymer’s purification. The obtained graft copolymers were used to prepare films via solution casting using chloroform. The –CH2Cl functional groups of the VBC grafts were then quantitatively converted to –CH2(CH3)3N+ quaternary ammonium groups via reaction with trimethylamine directly on the films, and the films, therefore, were investigated as anion exchange membranes (AEMs) for potential application in a water electrolyzer (WE). The membranes were extensively characterized to assess their thermal, mechanical, and ex situ electrochemical properties. They generally presented ionic conductivity comparable to or higher than that of a commercial benchmark as well as higher water uptake and hydrogen permeability. Interestingly, the styrene/VBC-grafted copolymer was found to be more mechanically resistant than the corresponding graft copolymer not containing the styrene component. For this reason, the copolymer g-VBC-5-co-Sty-16-Q with the best balance of mechanical, water uptake, and electrochemical properties was selected for a single-cell test in an AEM-WE.