Are saved games accessible from multiple devices?

@kelliwaugh  Saves are stored locally on your computer, not in Origin.  To transfer a save from one computer to another, you just need to copy over the file.  Saves are located in Documents\Electronic Arts\The Sims 4\saves, the Tray folder holds your saved households and builds (the contents of the in-game Library), if you want to transfer those too.

One thing to keep in mind is that a computer running the Legacy Edition will not be able to open saves that have been played with the standard version of the game, at least since the split happened in November.  Because the LE is essentially stuck at patch 1.56, if you try to open a save that's been played on a newer patch level, you'll just get an error.  However, saves played in the LE can be opened in the standard version.

If the laptop you were using was running the Legacy Edition, you should be able to play those saves on your current computer that also runs the LE.  Otherwise, if you want to play your old saves, you'll need to wait for a new computer that can run the standard version.