Hmm, that's very weird, since it's Unreal Engine 4 it does have XInput support out-of-the-box, I've even checked it locally on several gamepads and it worked. Some of the more rare controllers (mainly china-made) seemed to not work, but something like the XBox X Controller should work fine.
Do you have Steam on your machine? Can you verify it's listed below Controllers in Steam? Also, did you try to test the gamepad using the joy.cpl Windows utility?
Out of the top of my head, one thing to try is to actually enable explicit XBox Controllers support in steam - even if it's not a steam game, steam provides drivers for many gamepads, so it may help. Also, I would try to test it with any other XInput-supporting gamepad, just to be sure.
For instance, I've played through the game completely using the newly released 8BitDo SNK NeoGeo Controller, and it works just fine, both in wired and bluetooth modes. But for some reason, 8BitDo M30 Genesis gamepad does not.
