Remove the screws from the face of the heat vent grill. If the grill is painted on to the baseboard, cut the paint around the edges of the grill with a utility knife.
Remove the grill from the hole in the baseboard.
Measure the width and height of the hole in the baseboard. Cut a piece of plywood that is 2 inches wider and 2 inches taller than the hole.
Spread a bead of glue around the outside of one face of the plywood. Press the plywood against the baseboard, covering the hole, so it overhangs the hole by one inch on each side.
Drive a screw through the plywood and into the baseboard in each corner of the plywood.