I'm interested in doing some user research to see how documentation/help is being used in our enterprise web app, and I'm not sure what the most effective method would be.

I'm not looking to find out "can the user find X in the system's help" - that could be accomplished via some relatively straightforward usability studies. I'd like to find out what questions the users have, and if we're answering them or not. This will tell us what parts of the help are most valuable, and if they are actually helping the users.

So what kind of user research would be most effective here?

Bonus constraint: assume that I'm not able to make code changes to get some kind of "was this page helpful?" quantitative data.