how in the world could I have my mac chess program work with arduino to send/receive info for a chess playing robot, or is this approach feasible at all?
Probably not feasible at all. First, unless the Mac chess program has a means to send out it's moves to a serial port, how could anything wired to the Mac be able to duplicate the moves on a real chess board? So unless you can modify the Mac chess program or write your own or find another chess program that does communicate it's moves out a serial port, I can't see the project as possible.
If you pass that hurdle the project becomes just very difficult and very expensive, but not totally infeasible. 
Lefty
If you pass that hurdle the project becomes just very difficult and very expensive, but not totally infeasible.
A 3 axis CNC machine with a gripper could be used to move the pieces, and would not be terribly expensive or difficult to build, and certainly not all that difficult to program to move to the required column and row, to move up and down, or to grip and release.
The hardest part is defining and implementing the serial protocol to define what the Arduino-controlled CNC machine should do.
and would not be terribly expensive or difficult to build,
Not knowing the OP's budget and experiance level, I would have reservations about that statement.