There are no half-human half-ayakashi in Natsume's world right? Yeah, there are few stories about a human fell in love and be together with ayakashi, but that's it.
So his grandfather is a human. I read the wikipage and someone speculate it was boy who give manjuu to Reiko and always scold her. The boy come from another town. I'm not fully read all the chapters yet. Reiko died young mid (20s-early 30s) and only be together with her daughter that born out of wedlock (Natsume's mom for few years maybe). Either he died earlier or Reiko fled away from him or he abandoned Reiko.
The dragon statue said that Natsume look like this man he met before. So i speculate he is a exorcist that understands Reiko and nice to youkai.
Because of that dragon statue that I believe his grandpa is an exorcist
But it might be wandering exorcist
Because they(natsume's relative) doesn't have any hint about the guy
Is the grandpa a wandering exorcist? Or an ayakashi?
But I think it's more to the exorcist