Can I play LEGO® Batman™ 3: Beyond Gotham with a gamepad?

Yes.

LEGO® Batman™ 3: Beyond Gotham supports two categories of gamepads: Xbox 360 controllers (which require you to install a software driver) and other supported gamepads, listed below:

• Logitech Cordless RumblePad 2
• Logitech RumblePad 2 USB
• Logitech Dual Action
• Logitech F310
• Macally iShockX Game Controller
• Mad Catz PC USB wireless game pad
• Madcatz MP-8888 USB Joypad
• Mad Catz Game Controller
• Saitek P2600 Rumble Force Pad
• Saitek Dual Analog Pad
• Saitek P2900 Wireless Pad
• Thrustmaster dual analog 3.2
• Thrustmaster Dual Trigger 3-in-1
• Thrustmaster FireStorm Dual Power 2
• ThrustMaster 2 in 1 DT
• Thrustmaster vibrating gamepad
• WingMan Cordless Gamepad
• WingMan RumblePad
• PlayStation® 3 Controller (Sony)
• PlayStation® 4 Controller (Sony) Pre-August 2016 model. Product ID: CUH-ZCT1
• Eliminator AfterShock
  
• USB Joystick (PS2 to USB Convertor)
• AIRFLO
  
• P880
• PS Vibration Feedback Converter
• Rockfire PSX\USB Bridge
• Logic3 12 button vibration
• 4 axis 16 button joystick (PS2 to USB Convertor)
• T Mini
• Dual Analog Rumble Pad
• PC-Line 4-Axis,12-Button with POV
• Xterminator Digital Gamepad
• Cyborg Rumble Pad

If you have an Xbox 360 controller, see How do I use an Xbox 360-compatible controller with LEGO® Batman™ 3: Beyond Gotham? in these FAQs for a link to the software driver.