If I recall correctly (it's been a while since I've last tried to read JP Bookworm), the brackets are a direct translation of a alternate/bold quotation mark.
Japanese quotation marks look like this: 「」
However, there's a second form of quotation marks: 『』
So (with only the words translated) it'd read: 「I want to order a 『stylus』」
English also has two kinds of quotation marks (single and double quotes) and single quotes are sometimes used for quotations-within-quotations, but the single quotes don't stand out nearly as much as those bolded quotes.
I'm not a frequent reader of untranslated Japanese books, so I don't know if there's a common usage of those alternate quotation marks (I've only seen them in Bookworm, but that isn't saying much), but their usage in Bookworm is as others have described: Myne is not speaking the language of the world she's in.
Interesting, I notice that once other people in the world start using the words, the extra quotation marks (and the brackets by extension) disappear; I guess as a way of representing that they've now become part of the world's language.