Section 13 of the AGPL v3:

    "Notwithstanding any other provision of this License, if you modify the Program, your modified version must prominently offer all users interacting with it remotely through a computer network (if your version supports such interaction) an opportunity to receive the Corresponding Source of your version by providing access to the Corresponding Source from a network server at no charge, through some standard or customary means of facilitating copying of software."

So, if you make any modification to the program, you also have to "prominently offer all users interacting with it remotely through a computer network" [a copy of the source].

Which it doesn't currently do, so do you have to add a header to every request with a download link, or what? Without any clarification from the authors they can choose to interpret the license however they like.

Even clarification may not help, the vague term is still there in the license. A legal department may suggest you avoid such a license. Usual I am not a Lawyer disclaimer applies.