Yes—but no additional schematic wiring is required. The existing ESP32-DEVKIT-V1 has Bluetooth Classic built in, so the gamepad pairs directly with the ESP32’s radio; gamepad inputs are handled in firmware, typically using the Bluepad32 library.
The hardware priorities are adequate 3.3 V power and keeping copper/components clear of the ESP32 module’s antenna area. Which gamepad are you targeting—PS4, PS5, Xbox, Switch, or generic Bluetooth HID?