You’re getting thermal radiation and convection confused. The ISS has giant radiators because it’s a right pain in the ass to turn heat into thermal radiation, and it cannot rely on convection to cool things like you can here on earth. Turning thermal radiation into heat on the other hand is pretty trivial. Just don’t reflect it and it’ll turn into heat. These things aren’t transporting heat across distances. They are transporting thermal radiation across distances. That works as well in a vacuum – if not better – as it does on earth.

If thermal radiation doesn’t work in a vacuum, how is the sun heating anything up?