Remove the old strap(s) by unscrewing the screws holding the straps. If the straps are secured by rivets, drill them out with a power drill or pop them out with the flat side of a hammer. If they are not attached by screws or rivets, cut the straps off with scissors.
Cut a new strap the same length as the old strap. Fold each end of the strap in the shape of a triangle so that the sides of the strap meet in the the middle and the top tip of the triangle is the outside of the strap. For vinyl straps, boil them in a large pot before measuring them.
Cut a hole through the center of the triangle. Use a power drill or a screwdriver. Make the hole roughly the same size as the screw or rivet.
Clean the chair frame with soap and water. Repaint if needed.
Attach all horizontal straps that need to be replaced first and then attach all vertical straps, weaving the vertical straps through the horizontal straps.
Take your strap and place it over the frame. Put a screw or rivet into the hole in the triangle piece on one side and firmly attach the strap to the chair. Pull the strap tightly across the chair, wrapping it over the frame, and screw or rivet the other side. For vinyl straps, stretch the straps as much as possible (while they are still hot from boiling) across the frame before attaching them from one side to the other.
Wrap the strap around the frame twice and place a solid bead of epoxy glue along the seams for straps that are not attached by screws or rivets. Use a pair of pliers to hold the strap in place on the other side while you wrap the strap around the frame and glue it into place.