You want to allow for two ways for a user to add a completed game moves?
The simple answer is to contact the authors of the referenced site/application and license its use within your site.
They may already have a more advanced mechanism that would allow for the graphical user interaction that would output the moves.
Unless you want the graphical entry points as a way to set the various peices in a particular location versus playing out the game up to the point, it could be simpler to do. To allow a user to enter a move at a time would not guarantee that the entered moves are/were valid unless you check as part of the graphical entry mechanism which could significantly complicate the interface.