Use a table saw to rip the full plywood sheets to make the bottom and sides of the vanity. Measure and cut carefully. Sloppy cuts or inaccurate measurements will throw off the entire structure.
With the plywood cut to size lengthwise, use a circular saw to finish cutting out the individual panel pieces. Use a guide, such as a metal carpenter's square clamped to the wood, to ensure a straight cut.
The contemporary design for this vanity uses dado joints for the plywood instead of butt joints for a cleaner look. Dado joints also add strength to the overall piece. Line up the side panels side by side and clamp them securely together with long bar clamps. Also clamp them to the work surface below.
Measure and cut three plywood nailers to fit snugly between the side panels. Two will go at the top and bottom of the back of the vanity to attach it to the wall; the other will go at the top front of the vanity to support the countertop and sinks.
Use a drill and a pocket hole jig to drill pocket holes in the nailers. Pocket holes start in the side of a piece of wood and come out the cut end, making it possible to "toenail" the piece with hidden screws. Drill two pocket holes at each end of each nailer.
Attach the nailers to the side panels with pocket hole screws.
Use a miter saw to cut horizontal plywood rails to fit the top and bottom of the vanity front, plus three vertical stiles. Attach the rails and the two side stiles with wood glue and finish nails, covering the cut edges of wood.
Measure and cut plywood for the doors. Rip the wood stock on a table saw and use a circular saw to cut them to the correct height. Sand down the rough edges.
To install the vanity, remove the doors and set them aside. Drill pilot holes and drive screws through the nailers to attach the cabinet to the wall studs.