I own the Stellaris base game on Steam but have been waiting to pick up the DLC. If I purchase the DLC on GOG will it work with the Steam version of the game? Or do I need to repurchase the game entirely on GOG for it to work?

I believe technically it should work, but the number of hoops you'd need to jump through to buy, install, and keep both updated, make the effective answer, "no it won't work".

Pretty sure that, like on Steam, you need to own the base game on your service of choice to be able to buy any DLC

he asked if the DLC would work if he bought only on GOG-not if it would work on both. You'd need the base game regardless though.

no it will not work and you need the gog base game to make the dlc run... except if you play multi-player against someone else and they have dlc then you get to use the dlc for that game, even thou you don't own it

to be clear, multiplayer = only the host dlc counts