In Silent Hill 4: The Room, Henry Townshend may find one in a locked cell on the second floor of the Water Prison level. To access it, he must rotate the rooms on the second floor while doing the Bloody Bed Puzzle (for instructions, read the Bloody Bed Puzzle article). It will be on the bottom corner of the table.
The Stun Gun has a short range but it stuns enemies upon contact, making it easy to switch to another weapon and finish them off. In harder difficulties, the Stun Gun is almost worthless due to the higher health level of the enemies. Unlike Heather's, Henry's does not need recharging.
If Henry equips the stun gun in the room with the enormous version of Eileen Galvin'shead and zaps it, the eyes will briefly flash.