>run out of oil
lol no
You mean:
>run out of cheap oil
As others have pointed out, there are ways to make non-oil derived plastics but I don't think they have the ability to scale up to meet our present demand while still keeping it affordable.
In the long term, the price of plastic will probably go up. Not by much but by enough that the days of cheap plastic crap will come to an end, and you'll likely start seeing fewer things made out of plastic where appropriate substitutes (metal, wood, card, etc) can be made. It'll be difficult though and any substitution will likely result in increased prices for consumers.
Remember, plastic has two things primarily going for it that will be incredibly hard to replace with anything else:
>1. easily formed into whatever shape you want via extrusion moulding, injection moulding, or compression moulding
>2. light af, which helps massively reduce shipping costs due to reduced fuel usage. Imagine a standard 50" TV, it comes with a plastic back casing. Now imagine replacing that with very thin sheet metal, say 0.25mm. That TV suddenly gets a fair bit heavier - now imagine that extrapolated over an entire shipping container full of TVs.
Of course with my last point there's the argument you could use thin aluminium instead of steel for the same weight benefits but I'm not sure if aluminium that thin could have the same strength benefits as either plastic or steel, not without giving it a lot of ribbing reinforcement (think something along the lines of an isogrid autism waffle used in aerospace). But then you're back to the problems of increased cost from manufacturing such an item.
Sorry for the long post, I had an autistic train of thought going and it was fun to vomit it on a page.