I've heard some people don't write during the intermediate days of the festival (Chol HaMoed). Is this a formal prohibition or just a custom? What's the origin of this?
Not writing on Chol Hamo'ed is a formal prohibition, the original source of which is Mo'ed Kattan 18b, in the Mishna at the bottom of the page. Writing is considered a skilled craft (meleches uman) and is therefore prohibited on Chol Hamo'ed.
The Shulchan Aruch codifies this halacha in Orach Chaim 545:1.
The reason that only some people don't write is because there are several possible reasons to be lenient on this prohibition under certain circumstances. One of these is that our writing today is not done with as much care as the writing that was being discussed in the Mishna and is therefore not meleches uman (Rema ibid).