As I answered here, I think I can somewhat understand what Egnor is getting at. God is not a physical being - he is spirit. God is also not just any sort of spirit. God is invisible; he is before all things, and in him all things hold together (Colossians 1:15-17). God not only created the universe, he sustains it in existence at every moment in time. Therefore, in normal conditions, one cannot expect to sense God like sensing a rock, dog, human.
To take an analogy, God is not just a super-powerful character in the game. He is the Game Designer who created the game, sets its rules, in fact allows it to exist by continuously providing the computer with electricity. As a character in the game, you wouldn’t expect to be able to directly “sense” the Game Designer like you could sense other characters using your regular organs. Does this make sense?
But there are ways to interact with the Game Designer. First, as Coyne mentions, he could choose to use his creation to reveal himself in special ways. We see this with Moses and the burning bush. The bush is not part of God. Rather, God intervened in nature, causing the bush to behave in a special, unexpected way that imparts some information about him. However, this is not a normal occurrence in the game. In fact, the reason it works is precisely because it is an unusual, non-repeatable event.
(This is actually why the Incarnation is so incredible and one of the greatest mysteries of all time. Jesus is not just a man used to manifest God, like the burning bush. Jesus is God, the same God that is before all things and created the world, yet Jesus is also man! How could this happen? It is almost unfathomable. This is really one the most radical ideas of Christianity.)
A second way to “interact” with God is to use our reason to deduce things about him. This is what Egnor likely means by reason as an “organ.” To take our game analogy, while we cannot (under normal conditions) talk with the Game Designer directly, we can deduce things about the Designer from thinking about the general principles of the game that we discover. We find it odd that the game and its world can exist at all, and that it contains incredible order and variety. Even when the Game Designer doesn’t intervene specially, he still has set up the rules of the Game to accomplish his will.
Of course, as you pointed out, “reason” isn’t exactly the same as other organs like our heart, tongue, skin, or nose. I think Egnor must have been speaking metaphorically here. That being said, reason most likely has some physical component. No matter how reason works, it is possible to form philosophical arguments for the existence of God using our reason. That is how we “sense” God. It is not an immediate, intuitive sensing but an indirect one.