Well, first, they need to contact the company and ask for the source code, after they themselves have signed up, since the AGPL demands the code be available to users.

If the company responds in the negative (i.e., denying they're using mastodon or outright refusing to provide the source) or if the company fails to respond in a certain amount of time (like a week at least, since this is a new company), then you can start legal action.

It seems to me that because this organization could not find a link or a github making the source code public (which AGPL does not require), they decided to sue. They made no attempt to contact the company and ask, and they're not making any allegation the organization has outright refused. Instead, they've presumed such based on lack of a public download.