Seat belt Lawn Chair...
I still like the old-style aluminum folding lawn chairs-to me, as a bigger guy, they are much more comfortable than the new collapsible camp-style chairs being sold today. However, the old plastic or vinyl replacement webbing is nearly impossible to find, and if you are lucky enough to find it, it may be in no better condtion than what you are trying to replace. I had picked up a roll of seat belt webbing at the flea market for a few bucks, and after seeing handbags woven from it in magazines, I decided I might try to re-web one of my vintage chairs. The result is better than I expected, plus the webbing is stronger and much more comfortable than the plastic stuff. No pinched leg hairs or sticking to it in hot, humid weather! My guess is it will last forever as long as i keep it out of the rain.
If you can't get a roll of this stuff, troll your local "u-pull" junkyard for seat belts. I don't think they can sell them complete for safety reasons, so just cut the webbing with EMT shears or sharp scissors. You should be able to get it for a song. Pull the belts ALL THE WAY OUT and cut as close to the retracting reel as possible so you get as much of it as you can. I would estimate about four cars worth of belts should be more than enough to cover your chair. This stuff is super useful for lots of other things as well.
Look closely at your chair-if it still has the original webbing, examine how it is put on, it's pretty simple. Take a digital photo for reference, just in case you forget. You may want to replace the often-rusty screws with new ones also. Take your time and the results will surprise you. I also took a lighter and singed the cut ends so they would not fray, and singed the holes I drilled for the same reason. You can probably use a leather punch for holes as well. DON'T BURN YOURSELF ON THE HOT, SCORCHED WEBBING! Let it cool after singeing it, it's just like melted plastic until the temperature drops, and will stick to you like napalm.
For racing fans, you can attach a set of shoulder straps, complete with buckles to strap yourself in at races and to keep someone from stealing your chair when you get up! You can also use alternating colors, I just used this blue because I had a huge amount of it. Remember there is tan, gray, maroon or burgundy in addition to good old black for mixing and matching. I also remember the Pontiac Sunfire perhaps having bright red as well as an aqua "moon rock" color in the mid 90s (?) Do your favorite team colors or match the rest of your patio furniture. Black and Tan (for beer drinkers) would look good I think, as well as blue and Maroon (looks like the Gemini space program to me somehow). I think these lawn chairs are comfortable enough to use inside if company comes and you need extra seating, at Superbowl or holiday time.
You can also use this procedure for old, Mid-Century Modern-styled folding chaises and chairs-some of them were originally slung with webbing, or woven with macrame' type patterns and twine. A cheap flea-market wooden chaise or folding chair could yield a masterpiece... Go nuts, have fun.
UPDATE: A search of Ebay yielded new webbing in every color of the rainbow...it's not as cool to buy new if you ask me, but if you need to match with colors like orange or safety yellow, this is the way to go I guess.