>>313253975
That's interesting. Theoretically you can actually.
But your wishlist must be on point so your secret santa knows what kind of gift would fit you. Kind of like id you play pavlov then maybe a pavlov based christmas card or if you are beatsaber, a fully packed customized beatmap.

But in short, YES YOU CAN!. The e-mail is there for that to be possible to occur.
But you do need something identifying at least so you can be coordinated between other people. Even a throw away name or a random animu pic that you will use consistently while coordinating with the organizer.