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If I recall correctly, Calibre is licensed under the GPL. That means, among other things, that you can use it for whatever the heck you want. Yes, that includes tying it into an online page or service.
Pretty much the only way to break the license is to modify the program's source code and distribute the modified binary without the new source. But the online converter isn't distributing the software, so it's all good.
(Note that I'm making some assumptions here. I haven't actually visited the page to see what exactly it''s doing, so I'm just going off your description)
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