How do I clamp on thin parts without distorting them?

Use common sense here, thin parts are always a problem to clamp without distortion. If the part is thin enough to bow when clamped but will spring back to normal when unclamped, you can possibly clamp the part tight enough to penetrate the material with the grips, then loosen the vise and re-clamp the part not so tightly without moving it. The grips teeth will still be in the material, and you can machine the part with caution and common sense dictating the speeds and feeds.