Yes.
In fact, they are even free to take the software, modify it, and run it for their customers as a hosted service. The key difference, however, is that the license requires that they must also let their user's download the source code of their version.
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In "Why the Affero GPL" [0], the FSF specifically mentions:
> The GNU Affero GPL does not address the problem of Service as a Software Substitute (SaaSS).
(Here, "Service as a Software Substitute" is the FSF's term for what the rest of the world refers to as "Software as a Service". According to the FSF, it "defines the bad practice more clearly and says what is bad about it.")
Another of the FSF's articles, "Who does that server really serve?" [1], explains the problems with SaaS with regard to open-source software.
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[0]: https://www.gnu.org/licenses/why-affero-gpl.html
[1]: https://www.gnu.org/philosophy/who-does-that-server-really-s...