Is there a way to reorder my pinned icons using keyboard shortcuts or an external app? I don't have a mouse and I want to rearrange them in a specific order without using a mouse to drag!
Win 7 Pro.
There doesn't seem to be any built-in keyboard shortcuts for rearranging the taskbar icons. However, you can use an accessibility feature called MouseKeys to effectively accomplish the same thing, albeit with a bit more work.
To enable MouseKeys, open the Start menu, type in
In the new window that opens, check the "Turn on Mouse Keys" option and hit Apply. Or if you're trying to do this part without the mouse as well, hit Space to check that option (it should be active already, or move around with Tab if it isn't), and hit Alt+P to Apply the settings.
Now you can use your numeric keyboard to move the mouse pointer around. To move the taskbar icons around, move the pointer to the icon you want to move and hit 0 (this simulates pressing and holding the left mouse button). Then move the icon to the new location and hit . (to release the mouse button).
More information on how to use MouseKeys, including all the numpad keys and their functions, is available in this Microsoft help article.