Construct can export your project as a Linux desktop app based on the Chromium Embedded Framework. This is basically a copy of the Chrome browser engine (Chromi...
The Local Storage plugin saves to a database in the app data folder. You shouldn't sync the whole app data folder on Steam Cloud though - usually you just want a small amount of specifically chosen data per user. You can do that with the File System plugin and Steam Auto-cloud - see this documentation for more details.