I know such programs exist, because I went looking a while back.
Problems is, I didn't find anything that wasn't a complete headache to use.
Having said that, I seem to recall most video capture programs that gamers use can do screenshots as well.
(such as fraps, or Bandicam)
I don't think the screenshot features of the free versions of these programs has any real limitation (not sure though.) - They may possibly have watermarks or other restrictions, though I think it's mainly the video which is limited for the free versions.
Personally I also had an older computer with non-standard graphics drivers. It's technically a separate program (And I don't know if it still exists, or if there is an equivalent version for other hardware), but the driver included a utility called 'ati tray tools' (shows you how old that is). - Most of the features are graphics card driver stuff (graphics settings, overclocking, resolution changes, etc.), but it also allowed you to set a screenshot hotkey. (it was in the same part of the menu that allowed you to show a driver-level FPS counter)
So... That's 3 options I know of...
You could try one of those, see if it does what you want it to.