Can rubber and plastic be made with other oils besides fossil fuels?

I understand that all plastic and rubber in use today to include everything from zip-lock bags to tires and medical equipment is manufactured from fossil fuels. What if somebody tried to make these things out of something like canola oil? I'm asking this because at some point (hopefully within the next forty years) after we've told the oil companies to piss off and let us produce clean energy, we're still going to be reliant on these materials and I'm trying to figure out what the source will be then. Please just focus on the chemistry part of my question and not the piss off part. :P