If you want users to be able to store some level data etc on your servers, then you need to work with a database and for example send the data to it by calling a php script that does the data entry into the database.
If you want to store it locally without any sharing between users, you can use the localstorage. In this case it would make sense to combine the map editor and game itself into one app, since then it would use the same local storage.
If you want users to be able to share their creations you would need a database again, except if you are going for an inconvenient solution like users needing to copy and paste long strings of text into their game.
At least that are the options that I see.
Node Webkit has a file writing function, but it did not work for me, so meh... won't cover that here.