If "istemek" usually takes a full infinitive form of the verb, why is it "tanışmayı" here?
Because it's not in the immediate vicinity of the verb. If you separate the object from its verb, you have to use the accusative.
Put "çok" at the end, and the infinitive is fine: O kitabı yazan yazar ile tanışmak isterim çok.
The word order in the English translation is not conventional and sounds unnatural. The intensifier, "very much", should be placed closer to the verb.
"I would very much like to meet the writer who wrote that book".
Neden kitabı? Neden kitap yanlış burada?
Kıtap here is definite object, so it takes the accusative
"O kitabı yazan yazarla tanışmayı çok severim" is not accepted. Is it actually wrong?
Because "I would like" doesn't mean "I like". It means "I want".
What does yazan yazar mean exactly?
the writer who wrote
Is 'isterdim' also correct for 'I would like'?
İsterdim means i used to want