The critique of natural theology since Kant has been that God is not an object of possible experience. I think the claim involves a confusion of experience and sensation, and that experience is a more subtle concept that can include God, even apart from mystical or religious experience.
Start with the title of the post. When said without qualification, we’re talking about the sort of thing an employer is looking for, or a thing that comes from maturity. We mean there is some domain of action or thought that we’ve become acquainted with over time, and this acquaintance has given us a nose for what is important, a facility in dealing with it, and perhaps a narrative or two of how things work in this domain. We assume, in other words, that experience involves not just sensation but insight and the grasp of something significant. If you ask your uncle about his experience in Vietnam, you expect something out of him beyond a mere catalog of sense impressions. If he tells you nothing but “it was hot” you assume he wants to tell you the most significant thing about the place, or at least the most significant thing he is willing to talk about.
Experience, in other words, goes beyond mere awareness or sensation or consciousness. You can’t just grab a couch cushion and call it an experience of a cushion. “To experience the couch” sounds more like an advertising slogan because it means you’ve discovered something significant about becoming acquainted with it. The sensation has found a context under a hypothesis, insight, governing idea, coherent narrative, etc.
It follows that “to experience an object” can either mean to encounter it (a) as a part of the sensation, or (b) as part of the ordering idea. In natural theology, no one encounters God as (a) – and theologians even give many proofs for why this cannot be done. But they insist that God is experienced as (b), as a creator, a guarantor of rationality, a primary agent, etc.. Plantinga would add that God is not the only thing so experienced as a (b) that is not (a), but that this class also includes the extra-subjective existence of the world, other minds, etc..
Just Thomism 