<p>I want to write a word addin that does some computations and updates some ui whenever the user types something or moves the current insertion point. From looking at the MSDN docs, I don't see any obvious way such as an TextTyped event on the document or application objects.</p>

<p>Does anyone know if this is possible without polling the document?</p>

As you've probably discovered,
Word has events
, but they're for really coarse actions like a document open or a switch to another document.  I'm guessing MS did this intentionally to prevent a crappy macro from slowing down typing.
In short, there's no great way to do what you want.
A Word MVP confirms that in this thread.