I've been playing for a few hours and noticed that my wireless Xbox controller does not vibrate, but a wired one that I leave plugged in does. I've tried unplugging the other controller and play as usual but still no vibration when it should vibrate, for example when cleansing the fast travel spots?

Is there any solution for this?

I suppose you have already tried that wireless controller with other games and/or tried to plug it with an USB cable.