What is the object of transitive verb “have” here?
I intuitively understand the meaning of the phrase “have at it!” “What happened to you, but I cannot 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 doesn’t sound as weird as other objectless to have, such as ‘The president has in the office’?
What matters is that there is no one to blame for what happened there.
I believe that your analysis is mistaken: to have in this expression is intransitive. You should work hard to make this assertion.
Why is the verb have transitive? Argumentatively, I would argue that to have at is a phrasal verb, whose object is syntactically the object of the preposition at. What do some other phrasal verbs mean for look at?
So to have in this idiom is intransitive, and the imaginary object is expressed as the object of the prepositional complement at.