Way to change the webbing on this picnic chair

Depending on how closely you're going to try to match the original finish (or if close enough is close enough) you could use wood filler. This product is basically saw dust and glue. Most of them are stainable, so you could fuss with it to match your desk's finish.

This is an example, I am not affiliated in any way with any company in the link, just the first thing on google.

Replacing is probably most practical.

But if you really want to fix it, you need something to support the weight. Tape and glue aren't going to do it. I would find a length of strong pipe (check the plumbing section at a home improvement store) that will fit nicely either inside the leg (if it is hollow) or over top of it. Make the pipe fill/cover as much of the leg as possible. Then attach with tape, glue, bolts, whatever - just to stop it from sliding out.

If the pipe fits snugly it should hold up fairly well.