Here is the actual portion of the GNU AGPLv3 that is relevant:
> 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.
There are a few key things to note (I'm not a lawyer, but the idea of the GPL family of licenses is that they be written so that developers can also understand them):
* It only applies to people who have modified their version of the software, so if you just run code provided to you by someone else you don't need to give network users the source code.
* It uses the term "users" which would imply that if you state that someone external to your company is not permitted to use the service, they would not be considered a user (and thus you don't need to give them a copy of the source).