explorer.exe (Windows Explorer)
But it is not called, it already runs and if it captures the hotkey, it simply facilitates the action.
When you press Win+PrintScreen, then in Resource Monitor tool you can see that it is explorer.exe which saves screenshot file to its location. Moreover, if you exit explorer.exe using its Exit Explorer (hidden) menu command, this keyboard shortcut stops working.
Currently Explorer does not have any command line parameters to initiate the action.
But for example using PowerShell or NirCmd you can send the key combination using command line. Sequence of these 3 NirCmd calls works for me:
nircmd sendkey lwin down && nircmd sendkey printscreen press && nircmd sendkey lwin up
Oh, and when NirCmd made it into the answer, you can save the entire screenshot simply by its savescreenshot command:
nircmd savescreenshot d:\screenshot1.jpg
The above linked documentation says:
savescreenshot [filename] {x y width height}
save the screenshot of current screen into the specified image filename. The following file formats are supported: .bmp, .gif, .png, .jpg, .tiff
You can also specify *clipboard* instead of real filename if you want to save the screenshot into the clipboard.
This command only works on Windows XP/2003/Vista (GDI+ is required)
You can also specify 4 optional parameters - x, y, width, and height, if you want to save only a part of the screen.