If a person received a Don't Starve key from a friend, got it from a giveaway, or bought it from someone else, they can only use the version that they are given and cannot convert that key into additional codes for other platforms.
Your best, next step would be to ask the person who gifted you or sold you the Chrome key if they used their Steam key. If not, perhaps you can have it.
Your other option would be to purchase Don't Starve from Steam or from the Humble Store and you would be able to get access to both the Steam and Standalone versions of the game.