I would expect that an RV store might have the right sized item. I don't think you will be able to buy just the rubber or basket. But you will not have to buy the entire sink, either. What you are looking for is a "basket strainer assembly". In the box you will get a new basket and a new "cup" to receive the basket. New gaskets for where the assembly mounts to the sink will also be supplied. If you're really lucky, you can just drop in the new basket. Most likely, you will have to remove the entire assembly from the sink and replace it with the new one. An easy, uncomplicated job, requiring minimal tools.
Home depot might even have the right size basket strainer assembly, as it might be the same as used in small apartment efficiency setups or in bar sinks.