How is this normally handled for books? Can you download a book from a digital bookstore and print it out and have it available in your library just like that? Or have the files for users of that library?
Itch is self publishing, so actually there is no such thing as an Itch game. The developer simply chose to realease it on Itch. Maybe it is sold in other stores too. So, unless you have an explicit license known that allows things like putting it in a public library, asking for permission should be prudent and possibly legally required. Even for free games. Just because you can download it, does not mean you have rights to redistribute it or archive it.