Can we use whatever game engine? Also do we have to post it online, or can we just make it a download?
Per the rules on the Overview page:
That said, you'll have to "post" the game to itch.io to submit it -- I see no reason that can't be a web game, a MacOS downloadable game, Windows game, etc. The caveat being that Web games are generally more accessible and you're likely to get the most plays with something easy for people to try.