I was hoping to use vertex groups combined with a modifier but I couldn't get what I wanted with simple deform and vertex groups. I just want a quick and effective way to move faces along their normal's to add random thickness in selected areas.
For the object you have in the picture, you can add two displace modifiers with a noise or clouds texture. Set one to effect the Y direction and the other to effect the X direction. Make sure Space is set to global.
You might like the results better if you first made a flat, segmented rectangle shape, then use a displace in only the x axis (Local Space this time), subdivide it, solidify it, and then use a curve modifier to modify the object like below. You can mess around with different displacement textures and amounts to get the look you're after.
If you do not want to move the surface, click shift z. so it will only move in the x and y directions.