When indexing Chinese names, should I put the first name in the surname box, as they put their surname first?
Unless there is specific instruction, you will index it in the given/surname style. The surname is the last name of the person. Ex:
Given Name: Cuong Chi
The researcher will (should) be able to discern the name.
Yes, they put the surname first, but is this also true of the person who created the document that you're indexing?
I have some experience with this question, because Hungarian also puts the surname first. In general, on things like ship manifests or census enumerations, every name was written in the same order: all surnames first or all surnames last, regardless of the particular person's culture/language. There are exceptions, of course; I've seen ship manifests where the page started out surname-last, but then there was a Hungarian family, and from there on down everyone was entered surname-first.
When there aren't other names on the same document to help determine which part's which, you have to rely on knowledge of the language: "Janos Kovacs" definitely has the surname last, "Egerszegi Krisztina" definitely has the surname first. The fun begins when either part could be playing either role, such as with unmarked patronymics in Hungarian ("Péter Gábor"), or surnames as given names in English ("Shepard Chandler").
The other consideration is whether the name is correctly entered on the document. This is especially relevant to Chinese names in an English context, I think, because there isn't a shared name pool: unlike "Egerszegi" versus "Krisztina", to an English ear/eye, there's nothing that makes either of "Zhang" or "Wei" sound/look more like a surname than the other.