Petroleum, or Oil, although not the only energy source in the world of Fallout, was a cause of the Resource Wars in 2052. America found it necessary to use the army to secure pipelines and oil fields, meaning oil was still very important just before the Great War in 2077.
The world was known to be using nuclear cars and energy, "fission batteries", and micro fusion technology. There are several possible reasons that oil was a vital resource for a civilization that could be powered by nuclear energy. Oil would still be a cheaper alternative to nuclear energy, or used to power lighter or smaller vehicles, such as airplanes, once distilled into other forms such as kerosene. Oil is also required as a lubricant for nuclear powered machines, and for the production of plastics. One of the known oil companies is Poseidon.
Oil-based fuel may have been cheaper to use than nuclear fuel even up to the Great War. It's known that Europe and the Middle East collapsed after the middle east oil fields ran dry at the start of the Resource Wars. That means that both Europe and the Middle East's energy economies were based upon oil. Oil, being cheaper than nuclear power, was used until oil reserves ran dry. With their collective economies focused on oil, there was no revenue diverted to making the switch to nuclear power, or at the time nuclear powered cars were not available. That suggests high start up or costs for nuclear powered transit, at least when the Resource Wars began. In the real world it's currently cheaper to run cars on electricity but the cost of electric cars are so much higher that few people are currently using it, hydrogen powered cars exist but are expensive, and oil remains dominate. During the U.S. oil embargo electric cars existed in the United States but even extremely expensive gas was cheaper to use in new and existing cars. A similar situation could have taken place in the Fallout world as well.
The Resource Wars started over 20 years before the Great War. The Fallout games take place in the world devastated by the Great War and the ruins are a snapshot of technology at the time.
In the world of Fallout there are many abandoned nuclear powered cars, meaning that part of the transit fleet was no longer using oil. Fallout 3, Fallout: New Vegas, Fallout 4, and Fallout 76 have a large number of cars stripped down to their chassis, making it impossible to determine what type of fuel was expended to propel them. The presence of both gas stations and nuclear cars could mean that America was still in the process of nuclear conversion; still making oil important.
Some vehicles were known to be still using oil fuels, meaning some unknown problem existed that prevented 100% nuclear power. Planes seen in Fallout 3, Fallout: New Vegas, Fallout 4 and Fallout 76, and a few cars in Fallout: New Vegas and Fallout 76, still appear to have liquid fuel powered engines. The cars that are known to be powered by nuclear energy are very large and heavy; this is likely a design choice as in our real world, a generator the size of a refrigerator can power a large nuclear submarine, and a generator the size of a laptop computer can power a deep space probe for upwards of twenty-five years.
Synthetic oil, though used in limited applications during WWII wasn’t marketed until the 1960s. If in the world of Fallout synthetic oil was never mass produced they would still be using natural oil for lubricants. Cars, planes, factories and power armor would all rely on natural oil lubricants. Even if a large switch had been made to nuclear power, oil, as a lubricant, would still be crucial to the machine industry.
Oil is used to make many materials, including plastics. Combat armor used composite plastics, in conjunction with ceramics, as did the T-51 power armor and the Needler pistol and its ammunition. Factory working of clays and production of plastics would require the use of oil, not only for the factory equipment but also for the treating of clay in the forms of various chemicals. Other important plastics are Polycarbonate, which would be used in impact resistant plastics like riot shields and/or face masks. An even less suspecting, but perhaps the most crucial plastic to the Fallout story, is polytetrafluoroethylene or more commonly known as Teflon. Teflon is a chemically resistant, low friction, metal coating discovered by Roy Plunkett in 1938, 12 years before the timeline divergence. As Teflon is needed in the highly corrosive process of Refining Uranium for atomic weapons, the very sort of weapons used to destroy much of the Fallout world during the great war of 2077, this could be a contributing factor to oil depletion.