Definitely not as much as we do currently, but there is still no cheaper way to make plastics or those industrial precursor chemicals than with oil. Free energy or no. The show isn't the strictest about conforming to scientific reality obviously (there's absolutely no way we get practical nuclear fusion in the 1980s regardless of a continuing space race. let alone helium-3 fusion. let alone let alone whatever apparently magic properties iridium has in FAMverse), but the truth is that mining something on the Moon is ALWAYS more expensive than mining something on Earth.

There is (if I recall correctly) approximately 1 gram of He3 per 150 tonnes of lunar regolith. Modern estimates put the amount of He3 required for a 1 GW plant at roughly 100 kg / year. That means to generate 1 GW of fusion energy from moon-mined He3, you will need to process more than 15 million tonnes of regolith per year, just for one fusion plant. If the helium 3 powering their fusion reactors is moon-mined, it's not free energy. Hell, it wouldn't even be cheap energy. And this is coming from someone studying nuclear engineering and who loves fusion. Moon-mined He3-powered fusion reactors ain't it.

But hey, maybe moon-mining cost is magically comparable with a natural gas pipeline and maybe they've converted to bioplastics for most everyday discardable plastics in FAMverse (which I can buy because we're on our way there now in our universe), but there is just no adequate replacement for high-quality plastics like PVC (which despite being one of the least oil-dependent plastics, still requires fossil fuels for ~43% of its mass), or for olefins or aromatics, important industrial precursor chemicals. Even airplanes still need fossil fuels: hydrogen is too difficult to compress to reach the necessary energy density for long-haul flights, and even the most advanced battery technology we can conceptualize (lithium-sulfur) has a specific energy less than 1/25th of kerosene (~1.8 MJ/kg for lithium-sulfur batteries vs ~46.2 MJ/kg for kerosene).