Are passengers allowed by law to record the audio of their interactions with the US immigration officers at the airport when clearing the US immigration as they enter the US via air travel?
If state-specific, I'm mostly interested in California.
It is generally legal to voice-record under US and California law (in CA, the question could arise as to whether the agent has a reasonable expectation of privacy, but they don't when it comes to "public interactions"). But California law has to give way to federal law in the customs controlled zone in the airport, because border crossings are under federal jurisdiction.
It is not clear if there is a CBP rule/directive that prohibits voice recording The 9th district has ruled on a related question, where there is a First Amendment right to photograph – there is. However, that right varies depending on the "forum". In the case of "out in public" such as in a park, any content-based restrictions would be subject to strict scrutiny. But
In contrast, restrictions on speech in a nonpublic forum must only be “reasonable in light of the purpose served by the forum and view point neutral”
Given the tests for judging publicness, the airport customs zone is clearly not a "public forum". That means that any such restrictions only have to be "reasonable", and it is hard to fail rational analysis.
There is a no-cell-phone rule, but there are other ways to record voice; hence there may not be a rule that addresses Zoom recorders. In lieu of an explicit policy, you might be detained / moved to an interview room where a supervisor will determine what the rule is (and how that impinges on your desire to record).
There is a separate matter that CBP is supposed to record "incidents", so at a certain point your interactions will be recorded.