Remove the old webbing by taking out the hardware pieces -- usually screws or bolts -- that hold each strip in place on the chair frame. Save the hardware to use again.
Lay the old webbing out flat to use as a guide for cutting new webbing strips. Cut a new webbing strip for each old strip, making them precisely the same length.
Fold the end of one long webbing strip to make a pointed end, folding over first one corner and then the other corner to create a point with three layers of webbing. Poke a hole in the pointed end with the awl, making the hole about 1/2 inch from the point.
Wrap the prepared end of the webbing around the top frame of the chair to align the hole in the webbing with the hole in the chair frame. Insert a screw or bolt through the hole in the webbing into the hole in the chair frame and tighten it with the screwdriver or wrench.
Prepare the other end of the webbing by folding it into a pointed end and punching a hole through the end in the same fashion as the first corner.
Pull the webbing strip straight down the back of the chair, around the back of the rod that separates the back of the chair and the seat of the chair and across to the frame at the front of the chair seat. Fold the chair into the closed position to ensure that the strip will fit the chair properly in both the open and closed positions.
Wrap the webbing strip around the front frame and then under to align the hole in the webbing with the hole on the frame. Insert a screw or bolt to secure the webbing to the chair frame.
Repeat the same process to string each vertical webbing strip onto the chair.
Weave the top horizontal strip under and over the vertical strips to position it in place in the chair frame. Do not secure it to the chair yet. Weave the next horizontal strip over and then under the vertical strips to position it in place. Continue weaving the horizontal strips through the vertical strips until you place each one on the chair.
Prepare the end of the top horizontal webbing strip by folding it into a pointed end and punching a hole. Wrap the pointed end around the frame to the backside, aligning the holes precisely. Insert the screw or bolt into the holes and tighten it. Repeat the same process with the other end of the top strip. Repeat the same process with each horizontal strip.