I don't see any difference between using SMTP (email) versus HTTP (web) to submit inputs to the software. They are both application-level protocols used over a computer network; I don't see the license applying differently merely because of a change in network protocol.
One possible difference, though, is whether the user's input is supplied directly into the program or not. Requests submitted via either HTTP or SMTP could equivalently be (1) fed directly into the program or (2) placed in a queue, where they are eventually interpreted and handled by a human, who employs the program to produce (and possibly share) output. Admittedly, this second case sounds more likely for inputs supplied over email.
There are certainly cases of the second category that I would not expect to carry AGPL obligations. If you email a photo to a person and ask, "Please run this photo through your AGPL program that applies various color filters to images, and send me back your top three favorite outputs," then I think it's crystal-clear that the original person submitting the photo and receiving the three outputs back is not the person interacting with the AGPL photo-manipulation program. There are also cases that seem much closer to direct interaction: suppose the human in the middle doesn't apply any discretion or decision-making, but merely double-clicks on a script file twice a day to process an input queue. In such a case, it certainly sounds like the original submitter has a much more meaningful interaction with the program than the human who is merely acting as a biological cron job.
That said, there is (as far as I know) currently no case law that approaches the question of what does or does not constitute "interaction over a computer network" under the AGPL, so I cannot begin to supply a legal test of what would fall inside or outside this category. Frankly, it's not clear to me if invoking the program via an actual twice-daily cron job to consume network-supplied input, with no human involved, is sufficient to disconnect a user from "interaction" with the program. I can imagine different hypothetical standards that would include or exclude it, but no such standard exists yet in actual legal fact.
Also note that the AGPL's network-based source-sharing obligations in section 13 only come into effect "if you modify the Program," so if the company is only privately executing an unmodified copy of the program, they cannot have copyleft obligations, even if their case constitutes network interaction.