There are a few freeware tools out there (e.g., Taskbar Shuffle, XNeat, etc.) which have the ability to change the order of the buttons on the taskbar, without actually closing or opening any windows. Unfortunately, none of them appears to be open-source.
What are the API calls required to change the order of buttons in the taskbar?
A quick web search found this.
Seems to be exactly the style of program you're looking for, with the source available too.
I'll not go into more detail as it's explained pretty well here
The fact that the Windows API does not expose methods to rearrange taskbar buttons is intentional. There is no supported way to do this.
See this article (and the ones it links to) for the thinking behind why the shell developers don't expose this functionality.
However, resourceful people have devised hacks the accomplish this (see the other answers). I suspect these methods will fall apart as Windows evolves (Windows 7, 64bit, etc...). Don't be surprised when these techniques stop working.
The answer is to use TB_MOVEBUTTON in a call to SendMessage(), as described in the WinAPI documentation here.
The first parameter to SendMessage() (hWndControl) needs to be a reference to the toolbar that contains the taskbar buttons. This is non-trivial to retrieve, but the CodeProject entry referred to in Chris Clarke's answer has all the code required to retrieve this handle.
wParam and lParam need to be set to the button ID of the button to move, and the position to move it to, respectively. These IDs are the idCommand field in the TBBUTTON structure that represents each button; how to retrieve these structures for the buttons can also be taken from the above CodeProject entry.
> Seems to be exactly the style of program you're looking for
It's a complete bullshit (with hardcoded windows classes !!)
The rigth and official method is to use Win32 COM. See MSDN