You have a lot of choices. You could simply apply vinyl or even leather (if you are well heeled) or carpet (like they use on pro sound equipment) or fake fur.
If you want a flexible plastic, you can buy thin sheets of Kydex, which has what is called "hair cell" grain texture on one side. Kydex can be stretched or shaped after it is heated in the oven or with a heat gun.
I would think you could buy some 1/32" thick Kydex sheet, glue it on one side of the cabinet, wait for the glue to dry, heat the corner and fold it 90° for the next side of the cabinet.
Since Kydex is pretty soft it is easy to work with, cut, and shape. It is available in a number of colors and pretty durable.
However, you might just as well use vinyl and contact cement. You can even buy the same stuff that is used on guitar amps.