Do we need crude oil to make plastic?

No.  We'll find other ways to make things like plastic.  We already have, in some cases. We generally don't need to do it now, but it's technically reasonably straightforward, and affordable.

1) it's not made from crude oil - it's made with NGLs and natural gas, which are in surplus right now.

"In the United States, plastics are not made from crude oil. They are manufactured from hydrocarbon gas liquids (HGL) and natural gas. HGL are byproducts of petroleum refining and natural gas processing. These liquids are used as feedstocks by petrochemical manufacturers to make plastic and are used as fuels in the manufacturing process.  Hydrocarbon gas liquids (HGL):  A group of hydrocarbons including ethane, propane, normal butane, isobutane, and natural gasoline, and their associated olefins, including ethylene, propylene, butylene, and isobutylene. As marketed products, HGL represents all natural gas liquids (NGL) and olefins. EIA reports production of HGL from refineries (liquefied refinery gas, or LRG) and natural gas plants (natural gas plant liquids, or NGPL). Excludes liquefied natural gas (LNG)."

2) Plastic uses less than 3% of overall oil consumption.

In 20101, about 191 million barrels of HGL were used in the United States to make plastic products in the plastic materials and resins industry, which was equal to about 2.7% of total U.S. petroleum consumption. Of those 191 million barrels, 190 million barrels were used as feedstock and 1 million barrels were consumed as fuel to manufacture these products.

In addition to HGL, about 412 billion cubic feet (Bcf) of natural gas were used to make plastic materials and resins in 2010. This was equal to about 1.7% of total U.S. natural gas consumption. Of the 412 Bcf of natural gas, 13 Bcf were used as feedstock, and 399 Bcf were consumed as fuel to manufacture these products."

http://www.eia.gov/tools/faqs/faq.cfm?id=34&t=6

3) There are pretty good direct substitutes for many uses. For instance, a huge fraction of plastics are used for things like disposable beverage containers and food packaging. Those could easily be made out of aluminum, glass, and cardboard.

4) plastic is recyclable. We don't do a great job of it now, because we don't have to. If you recycle 95% of consumption (cars, for example, are 99% recycled), you only need to produce 5% as much.

5) We have enormous supplies of fossil hydrocarbons in the form of coal, methane, kerogen (shale oil), heavy oil, bitumen (tar sands), peat, etc, etc. for the next 200 years. They have problems for use as fuels: they don't flow like conventional oil in a way that prevents peak oil, and some have low or negative E-ROI, but they're enormous and perfectly affordable for materials like plastics, which are  niche, high value uses.

Materials like plastic don't have a CO2 emissions problem, assuming they're not burned (see recycling, below).

6) There's biomass, which is pretty badly suited for conversion to liquid fuel, but good for hydrocarbon feedstock.

7) Plastic consumption can be made more efficient by reducing packaging and redesigning structures to reduce density while maintaining strength (human bones are a good example: they're hollow, and even the tubular structures are mostly empty space internally).

Even now, much plastic is made from natural gas and coal.  Industrial chemistry can produce very simple hydrocarbons from any source of hydrocarbons, and build them into any compound our heart might desire. Various kinds of feedstocks would work. Some are more convenient or slightly cheaper than others. Whatever fossil fuel is convenient will work; biomass will work just fine, or hydrocarbons can be synthesized from seawater, atmospheric CO2 and renewable electricity (air, fire and water!).