Reading through a list of questions on a small questionnaire that came with a Japanese textbook, I found the following:
どんな本があったら買いたいと思いますか？
I am puzzled by the combination of the conditional of ある with かいたい.
どんな本が買いたいと思いますか？
Clearly means "what kind of book would you like to buy?". Normally I would expect an あったら to split the sentence into two clauses, but that would split どんな from 買いたい. What, grammatically is going on here?
I think the Q&A @user3856370 has pointed you to should sufficiently answer your question. I'd like to say a few words about how I personally understand that sentence. Here goes nothing:
Very often you see Japanese verbs used unaccompanied by a subject or an object, or both. See this answer for a more detailed explanation on this.
どんな本が買いたいと思いますか？
Your proposed version dispenses with the あったら conditional and has the patient of the verb どんな本 as the subject of the sentence. It is thus a single-clause sentence with どんな本 as its subject. Let's look at the original sentence:
どんな本があったら買いたいと思いますか？
You are right that あったら often marks a clause. In the original sentence, although there is no 読点 (comma), 「どんな本があったら」is 主節の述語を修飾する副詞節 (an adverbial clause that modifies the predicate of the main clause), because it describes a condition that, when met, leads to the next step (thinking about buying them).
If we put back in the sentence things that are omitted:
どんな本があったら、それを/が買いたいと思いますか？
What kinds of books, if the store has them, would you like to buy?
Since either が and を can be used here, it doesn't really matter if you regard the pronoun それ, a placeholder for どんな本, as a syntactic subject or object. Semantically of course, it is the object/patient of the verb 買う.
