"Can I get half of it?"
- after Particles follow the word they relate to, that's what they're treated together with.
を in particular marks the direct object, the thing that receives the action of the verb. It is usually a noun, and 半分 actually can be a noun. The problem here is that the sentence lacks context (and is in general lacking pretty much everything, kudasai in this sense isn't even an action as much as a "please"). Depending on context, it may as well be an answer to "how much do you want?", in which case it's determining an amount, which is neither a noun, nor an object.
In short, "half please" is a more accurate translation without giving way to confusion.
To add to other answer, another exception is words like "nanika", "nanimo", "dareka", "daremo", etc ... We treat them like words ("something", "nothing", "someone", etc) but they already kind of have a particle and do not require the particle "wo". Thus, you would say "nanika tsukurimasu ka?"
Sometimes they take other particles, and you should look up examples to see how to do this. One example is "darenimo", which is "nobody", but with the "ni" particle.
Not even "kind of", grammatically speaking, in Japanese, those are actual particles tacked to the question words: "mo" negates (nothing, nowhere, etc, used in negative sentences); "ka" for indefinite pronouns (somebody, sometime, etc); "demo" for "any-". This is why these words in particular go without additional particles in sentences.