I'm afraid I can't agree with the other answer to this question. The normal GPL-style obligations in the AGPL (ss4-6) are engaged by the conveyance of the software (all three sections read "You may convey... provided that..."). Both GPL and AGPL define (in s0) "conveying" as "any kind of propagation that enables other parties to make or receive copies", and "propagation" as including "distribution (with or without modification)".
Distributing copies within a corporation is therefore propagation, but doesn't constitute conveying and thus doesn't engage ss4-6, because the copies remain within a single legal entity (the corporation), and thus no other parties are involved. The GPL FAQ is clear on the subject, when it says:
Is making and using multiple copies within one organization or company “distribution”?
No, in that case the organization is just making the copies for itself.
This isn't any kind of "exemption" for internal use, it's simply an observation that because there's only one party involved (the company), no conveying has occurred, and thus ss4-6 are not engaged.
The language of AGPL s13 is very different. It says nothing about conveying; its requirements are engaged by the acceptance of an offer of interaction with a modified version over a computer network:
if you modify the Program, your modified version must prominently offer all users interacting with it remotely through a computer network ... an opportunity to receive the Corresponding Source of your version
This is unaffected by any limitations arising from the definition of conveying. The GPL FAQ also makes no mention of them, when it says:
A company is running a modified version of a program licensed under the GNU Affero GPL (AGPL) on a web site. Does the AGPL say they must release their modified sources?
The GNU Affero GPL requires that modified versions of the software offer all users interacting with it over a computer network an opportunity to receive the source. What the company is doing falls under that meaning
There is no similar language there "exempting" internal usage, or restricting its analysis to websites that are publicly-accessible. By your own admission, you are offering users interaction with "a modified version of a program licensed under the GNU Affero GPL on a web site". I see no reason to think that they cannot benefit from AGPL s13 simply because you work for the same company.