In theory, it should work, but I never had a chromebook, so can't say for sure. Since you said you uninstalled the game, and installed it again, it's not a problem that described in the instruction above.
To me, it sounds like the app doesn't have the permissions for the save data, I'd check that. But if that's not the case, I have no idea what it can be yet. Maybe there are some guides about Renpy games on chromebooks, but I haven't found any.