You don't need Ajax, you can use it if you want, but you don't need it..
I can't give you details on how it would work in your project, because I don't know about your project..
If the player can only place objects in a grid, you could either use an array or the tilemap-object to store the position and save that as json..
If the player can position anywhere, you might be better of saving each object as json..
If there are other limitations, you might be able to use those in the saving progress
I wouldn't save all the time, only when the player is ready and have the player enter a name for his map..
You could save the asjson to a localstorage item and have a dictionary object with the saved names..