Learn Croatian: Imate li karton za Anu Horvat? - Do you have the file for Ana Horvat?

In Croatian, feminine surnames do not decline. Because Ana is a woman, only her first name changes to the accusative case (Anu), while her surname (Horvat) stays exactly the same. If this were a man named Ivan Horvat, both names would decline (za Ivana Horvata).

Why do we use the accusative case (Anu) after the word za?

The preposition za takes the accusative case when it means 'for' a person or purpose. Since the file is for Ana, her name changes to Anu.

Does karton really mean 'file'? I thought it meant 'cardboard'.

You are right that karton usually means 'cardboard'. However, in administrative or medical contexts, it is the standard word for a patient's medical record or a physical file folder, because these records used to be kept on thick card stock.

Could I ask this question without using the particle li?

Yes, in everyday spoken Croatian you can just say Imate karton za Anu Horvat? and make your voice go up at the end. Putting the verb first followed by li (Imate li) is just the more standard, slightly more formal way to ask a yes/no question.