After fiddling with this for quite some time, it's done! You can now play chess in Besiege from the keyboard!
Harnessing the power of the automatron, I programmed 18 of them to move the grabber according to keyboard keys corresponding to the chess table, from A to H and 1 to 8, then grab a piece using Enter and place it back down using Backspace.
Had to add 12 textures into the Workshop, since we have a peon, rook, knight, bishop, king, and queen to deal with, one set black and the other white. They are textures for rockets set at 0.1 fuses and explosion power, which is the way I managed to simulate "taking" a piece from your opponent: it blows up!
Haven't figured a way to simulate a peon reaching the far end of the table and turning into another piece... oh well, maybe in Chess Machine v2.0?
Link to machine: http://steamcommunity.com/sharedfiles/filedetails/?id=930074485
Building Tools http://forum.spiderlinggames.co.uk/index.php?threads/tgyd-s-building-tools-spaar-s-mod-loader-for-v0-42b.2806/
Easy Scale http://forum.spiderlinggames.co.uk/index.php?threads/easy-scale-mod-besiege-v0-42.3314/
No Bounds http://forum.spiderlinggames.co.uk/index.php?threads/nobounds-mod-besiege-0-32.1080/
Song: Dreamy Flashback by Kevin MacLeod