Interesting question - even after a few reads of the AGPLv3.0, I do not see where it requires you to make the source code available if you run an unmodified copy of the software on your server.
However, the FSF's page about the motivation to use the AGPL, Why the Affero GPL, contains the following claim:
If some program on this server is released under the GNU Affero GPL,
the server is required to offer the users the corresponding source of
This is embedded in a section about SaaSS, Software as a Service Substitute, which outlines the FSF's view on services that operate on data provided by users, but are outside of their control. The AGPL is said to not address this issue.
In its Basic Permission section, the AGPL states:
You may make, run and propagate covered works that you do not convey,
without conditions so long as your license otherwise remains in force.
"Propagate" and "convey" are terms that the FSF has invented to avoid ambiguities with copyright law terms of some jurisdictions. "conveying" software is what actually requires you to make source code available - but a user interaction with the software via a network, as in your question, is explicitly declared as not conveying:
To "convey" a work means any kind of propagation that enables other
parties to make or receive copies. Mere interaction with a user
through a computer network, with no transfer of a copy, is not
So yeah, at this time I am siding with "you do not have to provide the source code", but this is in spite of the claim that you have to on the FSF's own "Why AGPL" page. I hope someone can point out my error in a comment.