ID claims that something irreducibly complex cannot arise through natural processes so doesn't that imply that the designer is acting supernaturally.
It has a choice: it switches between what you say and the claim that something irreducibly complex cannot arise through known processes. It then does a bait-and-switch game to replace "known" with "natural", so that it can invoke the supernatural designer. The end result is an excellent example of the fallacy of false dichotomy: ID proponents try to make it seem like if you don't accept every jot and tittle of evolutionary theory, you must then accept their answer of deliberate design by an intelligent and supernatural creator entity. They gloss over the fact that there are a number of other possible explanations, such as an unknown but natural mechanism, a supernatural but unintentional cause, or deliberate design by a stupid supernatural entity.
Of course, all this is moot until someone comes up with a valid case of irreducible complexity, which hasn't happened yet.