Yes. To achieve this you should light the apparatus with infrared light using something like the ANY-maze infrared illuminator. Provided your camera doesn’t have an infrared blocking filter then the video picture will look as if the animal is in normal light. A word of warning though: many webcams do include infrared blocking filters, so you need to check that first. Alternatively use the ANY-maze USB camera which doesn’t block infrared and works very well in this type of situation.
A handy tip in this situation is to point the infrared illuminator at the ceiling and let the light reflect down onto the apparatus – this lights the apparatus more evenly than would occur if you pointed the illuminator directly at the apparatus (which tends to light the middle more than the edges – like a flashlight would).