You will need to be able to lift pieces, rather than just slide them, for example to support knight moves and perhaps castling.
And, because you need this capability, you might as well build that as the main way of moving pieces.
This means you need three axes, and a gripper.
This is not particularly different from a "pick and place" machine that populates electronic circuit boards with components. Maybe that gives you some search terms.
Also, the bigger you make something, the more expensive it becomes, by a factor that I'd estimate to the square of the volume supported. For a bot that can do 4 feet one way (1.2 m) and 8 feet the other way (2.4m) with a third axis, but no gripper, look at the ShopBot PRS Alpha CNC router. It's a $15,000 class machine.