There is a 'red rubber' that is used in industrial maintenance as gasket replacement, it has more stiffness than black rubber, yet is still piable. About 1/8 of an inch thick. I replaced a Wing (round, 6 arrow) rubber gripper with it over thirty years ago, it is still fine today. Still holds arrows tight and has no dry-rot.
Be sure that you cut the arrow hole and opening under size, so that the shafts have a snug fit when they snap in. I used the old gripper as a template and a piece of arrow shaft to punch the holes.