As far as I can tell the game does not support the GOG Galaxy overlay, so F12 key does not work for screenshots.
I don't think the game has a dedicated key for screenshots, either, but I could be wrong -- I couldn't find any documentation on this.

I'm on Windows 10 x64, on a multiple monitor setup, and sometimes Print Scren key works for me, sometimes it doesn't and I get a black screen. It's not related to what happens in-game, I think, but I haven't figured out what makes it work.

An alternative working solution is that if you have a Steam client, then you can add the game in your Steam library with "Add a non-Steam game to my library", and because Steam's overlay works with the game, you will be able to capture screenshots with Steam's screenshot hotkey (I've tested this and it works for me).

I also tested the classic legacy edition with ScummVM (latest version 2.5.1 and also a daily development build 2.6.0git). Alt+S works for me -- and it can be remapped to something else via the keymapper tab which is the latest ScummVM versions. The screenshots are stored (in Windows) under "This PC > Pictures" in a folder "ScummVM Screenshots". The full path for me looks like: "C:\Users\<username>\Pictures\ScummVM Screenshots" , where <username> is your windows login name.

The legacy edition in GOG is bundled with an older version of ScummVM (it seems to be 2.0.0) so maybe this could be a reason for Alt+S not working? In any case, a recent development build (2.6.0git) ScummVM version is recommended since there's been improvements to the Groovie engine (The 7th guest, The 11th Hour, Clandestiny, TLC) especially during the last year.