The Steam version of Dream certainly supports screenshots via the Steam API. But I purchased the GoG version. And I can't find a working keybinding to have the game store screenshots. Would I have to manually add one to the UDK INI files?

If you enable the "typing line" (or even the full "console"), you can execute the command "shot" to get a screenshot as BMP in Dream\UDKGame\Screenshots\Win32; unfortunately I was not yet able to bind this command to a key. And I might prefer the PNG format if it is supported...

Calling the screenshot commands in the full console will take a screenshot with the console, so you will prefer the typing line. There is also a high-resolution "tiledshot" command, but the result is not satisfying when the menu is displayed (it will be tiled in the overlay).

Not all keys seem to be recognized, here are some working examples:

Dream\UDKGame\Config\UDKInput.ini

[Engine.Console]
ConsoleKey=Divide
TypeKey=Insert