How does SpaceX plan to supply or recycle respiratory gasses?
As with most questions about SpaceX, the answer to this question is essentially that SpaceX is a private company, and unless either SpaceX or Elon Musk have made a public statement about this, the answer (IFF there is an answer at all, i.e. if they even have a plan), we simply do not know and cannot know.
In fact, even if there was a plan, and even if they publicly announced it, we have seen dozens and dozens of times that their plans change – and this is not an exaggeration – quite literally on a daily basis.
We can, however, make educated guesses. Elon Musk recently gave a two-hour long interview and tour to Tim Dodd (aka Everyday Astronaut), and over the course of this interview, he repeatedly said that at the moment, all their efforts are focused on how to get to orbit as fast as possible, since they have tested everything they can test so far, and really need to go to orbit to achieve further test objectives such as refining the heat shield, testing the hypersonic, supersonic, and transonic sections of launch and reentry, testing the stage separation "non-system", and so on.
At one point in the interview, Elon Musk even seemed annoyed about a question, when Tim Dodd asked him two consecutive questions about the oil rigs, and he answered both with "We're not thinking about it."
So, my guess is that the answer to your question
How does SpaceX plan to supply or recycle respiratory gasses?
is "They're not thinking about it".
And by that I don't mean that they don't plan to keep their astronauts alive (obviously), I mean that they simply ignore the problem until and unless it needs to be solved.
We have seen this a couple of times already:
- Landing legs: it was clear that the landing legs used on Starhopper, Starship Mk1, Mk2, SN1, 3–6, 8–11, and 15–16 were always just placeholders, designed to do the minimal thing required for testing a single landing. And people constantly asked "When are they going to start designing the real thing?" – and in the end it turns out, they simply designed them away. If they had invested significant effort into designing the landing legs, all of that effort would have been wasted. Obviously, they will still need landing legs for the first couple of flights to the Moon and Mars, until they can install catching towers there, but those those can actually be significantly simpler and lighter than the ones needed on Earth due to the lower gravity.
- Stage separation system: there had been questions about this. But until now, they never needed to separate stages, so why waste time in designing one? And in the end, it turns out, they don't need a stage separation system.
- Hot-gas thrusters: they have been part of many official renders, and were mentioned by Elon Musk many times, but it took very long for them to appear in the form of fit check test articles. They were installed for a short time, then removed again, because they are actually not needed for the currently planned flight profile. And now, it turns out that using ullage gas venting for reaction control may very possibly be enough, so they don't actually need the hot-gas thrusters at all.
- Lunar landing engines: they are shown on the official NASA and SpaceX renders, but when Tim Dodd asked Elon Musk about them, he said they are not even sure they need them at all. Depending on how exactly the landing profile of Starship turns out, and how exactly Lunar regolith reacts, landing on a single throttled down Raptor engine may actually be fine. Again, if it turns out they don't need them, why would they design them?
So, while I am sure SpaceX does not want their astronauts to suffocate, I find it plausible that "they're not thinking about that" until they actually get ready to fly a hundred astronauts for a period of months. Right now, the longest flight they did was about 8 minutes, was within breathable atmosphere the whole time, and had no astronaut on board anyway.