Plastic, I would say, is among the least of their worries. By the time they land on Cleft, they have limited amounts of all types of consumables, especially water and food. In the extended short term, they can manage to sustain a very small population through a very efficient recycling process, but the fact remains that Izzy and the Arklets brought effectively jack squat when you look at the prospect of building an advanced society in the hundreds of millions.
[by the way, I've always understood Cleft to be (part of) the solid core of the Moon; if so, they might be short-strapped for metal variety as well, given how the solid core of the Moon is mostly a large chunk of iron with tiny deposits of nickel and sulfur]
There are some clues in Part 3 of the book. There are a couple of places where they mention how, after the Hard Rain started to abate, they replenished the oceans by altering the orbits of ice comets. This hints that they retained a notable knowledge about space travel and civil engineering. It is not a stretch to think that they used the same technology to harvest things they needed (water, metals other than iron, certain naturally-occurring compounds, etc) from elsewhere in the Solar System.
With respect to your specific question, plastic is ubiquitous in our current world largely because it is cheap and easy to produce in very large quantities. I'm looking around my office right now and I really can't find that many things made out of plastic that would lose their functionality if they were built out of something else. A society with only small amounts of plastic would be different, but not necessarily worse, let alone dysfunctional.