Right click on All Programs from Start Menu and chose Open all users. Then double click Accessories (the path is also here: C:\ProgramData\Microsoft\Windows\Start Menu\Programs). Drag the Snipping Tool from your Desktop into that Folder.
Or like Mickey said, drag and hold down on the Snipping Tool from the desktop, and drag it over to the Start Menu button, it takes a few seconds, then hover over over All Programs, then hover over Accessories. Then drag in there. But after hovering it will take a couple of seconds to Open.