Local, Global, Normal hotkey

I’ve been looking through the manual and searching online to try to find out if there is a hotkey for quickly switching between the Global, Local or Normal of a given oject/edge. thanks.

No, and that’s because nothing but Texture Mapping and some parent heirarchies work as either/or. Though, if the answer were yes, then the hotkey would be Tab. In Object mode all objects (including their geometry and datablocks) are translated in Global space. In Edit Mode only the geometry of the Active Object (the one selected last) are translated in Local Space (relative to the Object Center).

Sorry, just realized you are wanting a hotkey to the widget icons. No, as in the transform panel you need to select the mode with a click.