Tyvek is made of high-density polyethylene (HDPE) - that's the same material as shampoo and detergent bottles are made of. Therefore Tyvek bibs can be recycled as HDPE 2 class waste into a variety of different products, provided that they have no additional materials attached to them, e.g. foam from timing tags etc.
If you want to recycle your Tyvek bibs, get in touch with your local authority to confirm that they do accept HDPE 2 class recycling waste or contact your race bib provider, who might be able to help. Additionally, you can get in touch directly with DuPont's Tyvek recycling program.