Old inner tubes, found at auto or tire recycling yards, can work as rubber sheets. These are about 1/16 of an inch thick, and fairly stiff. Common scissors cut the inner tube apart. If you need extra thick rubber, look to old non-steel belted tires. Tires are hard to cut apart, so use a saber saw or a hacksaw blade to hand cut. Bicycle inner tubes are another source of sheeting. If you need a rubber tube, bicycle inner tubes come in various diameters.
Mouse pads are really a rubber sheet with a cloth glued on. These are fairly soft, so it will "squish" when compressed under a load. Mouse pads can also be cut with common scissors. The only drawback is the size is about 6 inches by 6 inches. If you need a larger sheet, you have to somehow attach several pads together.
If you need thin rubber sheeting, look to children's balloons. These are very thin, about 1/64 of an inch thick. This rubber is highly stretchable and cuts easily. Balloons come in a myriad of shapes and sizes, so look for the ones that allow some oversize, so you can trim to fit.
Rubber gloves are also a good source for small sheeting pieces. Gloves come in various thicknesses and various strength factors. A pair of scissors cuts them easily. Furthermore, the fingers are premolded, so if you need a thin rubber tube, use the fingers. They are inexpensive enough so if you make a mistake, you don't have to worry about it.
If you need a large quantity, supply houses have sheeting by the yard. One such house is the Gooding Rubber company, which supplies sheets synthetic rubber. The problem with supply houses is sometimes they want an exact specification. They are staffed by engineers who sell to other engineers and speak using unfamiliar terms. For example, Gooding sells sheets made of nitrile or EPDM, in different durometer readings. Familiarize yourself with all the rubber sheeting terminology and jargon before approaching them.