Soft plastics
My first choice for soft jaws is UHMW Poly. This is the heaviest poly that is generally available. It is usually milky white.
Some cutting boards are made from this, you can tell because they are heavy, and feel kind of waxy to the touch. They are good for cutting boards because they will not chip. UHMW machines very easily, and can be drilled and tapped. There is no glue for poly materials, the only way to join poly is by plastic welding.
Next in the list for me is WOOD. Softer wood that is. Such as common pine or basswood.
PVC, same as the pipes, can be bought in rounds and flats. It is not as soft or resilient as UHMW poly. However, it can be easily glued and machines well.
Nylon is also good, but more expensive.
I have bought surplus plastics (cutoffs, or as they are called, 'drops') from:
www.machinistmaterials.com
Hope this helps
ADDED: Some thin, high density, rubber foam covered with fabric on one side can be used as a pad on the jaws. Where do you get such a material? Insoles for shoes!