Mark the center of the wood plug with a marking pen or carpenters pencil.
Equip a power drill with a paddle bit. A paddle bit is a flat drill bit with a point in the middle, used to create holes in wood. Measure the wood plug and match the flat surface size of the paddle bit to the plug.
Place the center pointed area of the paddle bit on the center mark of the wood plug.
Turn on the drill to grind the wood plug out of the hole. Drill slowly so as not to damage the chair or the fastener under the plug. Damaging screw heads under plugs will make removing them difficult.
Place a thin chisel or flat-blade screwdriver onto the center of the wood plug.
Tap the chisel with a mallet or lightweight hammer to crack and splinter the plug.
Rotate the chisel 45 degrees and tap again with a hammer. Keep the chisel positioned straight. Angling it can damage the chair.
Pull out splintered pieces of wood with needle-nose pliers. Chip away at the wood plug until the hole is empty.