I own Don't Starve for Steam. Do you know if I can install an expansion I bought on GoG? Or do I have to buy the base game on GoG too?

Go here, search for "Don't starve", try to follow the instructions.

You won't be able to buy a DLC here unless you: A) already own the base game here (or buy it at the same time); or, B) purchase the expansion as a gift code (which you will then be unable to redeem to your account unless it contains the base game).