Remove the cabinet doors from their hinges using a screwdriver. Take them outside to an work area where you can install the plexiglass.
Inspect the panels on the front of the cabinet doors. Some are glued in place. With a bit of careful prying with a putty knife, you can remove the panel without any cutting. Others are cut to look like a raised or inset panel, and these must be cut with a jigsaw.
Drill a hole into the bottom edge of the panel, making sure not to drill into the outer frame of the cabinet door. Slide a jigsaw blade through the hole and slowly cut out the panel, keeping straight lines along the edges. Remove the inside panel.
Sand the inside of the opening with 120-grit sandpaper to clean up any rough edges. Turn the cabinet face down on a flat work surface.
Cut a piece of plexiglass to cover the opening. Run a bead of silicone caulk around the edge and press the acrylic pane in place.
Hold the pane in place using mirror-mounting clips. Mark their placement with a pencil and drill pilot holes into the cabinet to keep from cracking the wood. Mark the depth of the screws on your drill bit with a piece of masking tape. Screw the clips in place and tighten gently.
Attach the cabinet doors to the base hinges.