I'm wondering which is grammatically correct:
Neither they nor I am an idiot at all.
Neither they nor I are idiots at all.
The US Navy has a term, blivet: 10 pounds of shit in a five-pound bag.†
You're trying to cram too much into a construction that isn't designed to handle the overload. When you get boxed in to problems like this, where every answer you can think of sounds wrong, Great Mother English is sending you a message: find another way to do it. For instance:
We're none of us idiots, neither me nor them.
They're not idiots, and neither am I.
† Blivet is also the name for this:
This "optical illusion that displays two irreconcilable perspectives at once" has been adopted as the name of a Python storage management module because it reflects the unhappy fact that "the goal of clean, consistent design is virtually irreconcilable with the tangle of inconsistent storage technologies that must be managed."
English Grammar rules are simple and straightforward. If subjects or subject and an object are separated by 'and' then verb will be in plural i.e. 'are'. On the other hand, if they are separated by 'or' then verb will be used for the subject or object following the 'or'. e.g.,
Neither they nor I am an idiot at all. (Right)
Neither they nor I are idiots at all. (wrong)
Neither I nor they are idiots at all. (Right but seldom used as the third person is iterated prior to first/second person)
Neither they nor you are an idiot at all. (right)
The second one is correct; you can't rely on "I" all by itself to tell you what form to use. Instead, "Neither they nor I" is what you're referring to, and that's clearly plural.
(Relying on this, though, is not wise: avoid any ambiguities by rewriting so this particular confusion can't occur.)