GPL distribution obligations do not ordinarily arise solely because software runs on a server. AGPL section 13 addresses a different event: users interacting remotely with a modified version of the program, where that version supports such interaction.
The source offer is to the interacting users. Do not substitute a claim that every dependent HTTP client must be published. Determine the scope of the covered program and any combined work from the supplied facts.
Internal-only deployment does not by itself answer the AGPL network question. Employees can be interacting users too. Conversely, the mere presence of an unmodified AGPL executable on a machine does not make every local program part of it.
Here the modified library and app are expressly one combined network service. The app owner authorizes compatible release terms. Repair the release terms and source offers for this modeled combined work.