Make three rectangle frames out of six planks of 36-inch-long, 1-by-2 inch lumber and six planks of 22-inch-long, 1-by-2 inch lumber. Make one frame from two more planks of 22-inch-long, 1-by-2 inch lumber and two planks of 32-inch-long, 1-by-2 inch lumber. All the 22-inch planks should adjoin the inside edges of the 36- and 32-inch planks to form the corners of the frames.
Secure the frames together with two 2-inch nails hammered through the longer side of the frames into the ends of the 22-inch side. You will have three frames that measure 24-by-36 inches and a single frame that measures 24-by-34 inches.
Lay out all the frames flat on the work surface and place a piece of 24-by-36-inch plywood on top of the corresponding size frame and a piece of 24-by-34-inch plywood over the frame of that size. Secure in place with nails every 6 inches around the edge of the plywood into the frames.
Keep one of the three 24-by-36-inch frames flat, and stand the others up on each long side of the flat frame, with the plywood side facing outward. The upright frames should be perpendicular to the flat frame. Secure in place with 2 1/2-inch screws driven every 6 inches through the bottom of the side frames into the sides of the flat frame.
Insert the remaining, smaller frame in between the upright frames on top of the flat frame. The plywood side of the frame should be facing out and be flush with the ends of the side frames. Secure in place with screws driven every 6 inches through the side of the side frames into the smaller frame, which will finish the three-sided box of the lawn cart.
Position a bicycle wheel on the side of the lawn cart box so that the axle of the wheel is around 16 inches from the closed back of the box and the bottom of the cart is 5 inches off the ground. Make a mark at this position, and a parallel mark on the opposite side of the cart box.
Mark the middle of two pieces of 5-inch-long, 1-by-2 lumber and place the lumber pieces over the marks made on the side of the lawn cart box, so that the marks match up. Secure the lumber to the sides of the cart with nails through the corners of each lumber piece.
Drill a pilot hole sized to fit 4-inch-long bolts through the mark on the 5-inch lumber right through to the inside of the cart box.
Insert a 4-inch bolt through the axle of the wheel and then through the holes in the cart. Tighten the nuts of the bolts on the inside of the cart to lock the bicycle wheels in place.
Place two 2-inch-diameter dowels that are 30-inches long on either side of the open front of the cart, around 2 inches from the top. Leave about 24 inches of dowel past the open front of the cart and secure in place with screws driven every 2 inches through the dowel into the side of the cart for handles.