Have an assistant hold up two 3/4-by-18-by-36-inch plywood boards parallel to each other on their longer, thinner surfaces. Keep them about two feet apart to be the side boards. Set a 3/4-by-18-by-24-inch plywood board against one set of their ends to be the top board.
Position 1-inch metal L-brackets between the top board and side boards, spaced every six inches. Mark the screw holes. Remove the brackets. Drill out the holes. Attach the brackets in place with 3/4-inch wood screws. Repeat with an additional 3/4-by-18-by-24-inch plywood board on the other end of the side boards to be the bottom board.
Set out a 3/4-by-24-by-37 1/2-inch plywood board on the ground to be the rear board of the vanity. Position the rectangular frame on the rear board. Arrange 1-inch metal L-brackets every six inches within the frame against the rear board. Mark the screw holes, drill them out, and attach the brackets with 3/4-inch wood screws.
Set another 3/4-by-24-by-37 1/2-inch plywood board on top of the frame to be the door. Put two side-mount cabinet hinges against the edge of the side board and the door board, evenly spaced to bear the weight of the door. Mark the screw holes of the hinges. Drill out the holes. Install the hinges on the door and side board with 3/4-inch wood screws.
Hold a magnetic two-part latch against the door and side board of the vanity and mark the screw holes on both. Drill the holes out and install the latch on each surface with 3/4-inch wood screws.
Drill a hole into each corner of the bottom board, up and into the side boards. Attach a screw-in furniture leg in each hole by rotating the leg until it is tight.
Position the vanity in place and mark the spots on the rear board and top board that must be cut out for the pipes and sink. Cut them out with a saw and make adjustments until they fit perfectly.