I intuitively understand the meaning of the phrase "have at it!", but I can't explain it to myself. I understand that "to have" in this sense requires an object to be valid, so why is it missing here yet it doesn't sound as weird as other objectless to have, such as "The president has in the office"?
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I believe that your analysis is mistaken: to have in this expression is intransitive.
Of course the verb have is normally transitive, which is why you assume that it must be transitive here. But I would argue that the to have at is a phrasal verb, whose object is syntactically the object of the preposition at. This is similar to a multitude of other phrasal verbs, such as to look at.
So to have in this idiom is intransitive, and the notional object is expressed as the object of the prepositional complement at.