On our team we hardly ever fabricate parts in a shop. So we don’t have a need for this but I believe Cheesy Parts is an option that you might be looking for.
It looks to be web based and very well made (Like everything from the Cheesy Poofs). I would recommend you check it out.
Our team is planning to use CheesyParts with a few modifications (to make connections to drawings) for our next season.
Prior to moving to a web based solution, we used a physical binder that logged the fabrication state of each specific part and contained drawings for each part.