A Nativity scene is a possibility to anonymously give gifts in a larger group.
You need a (wooden?) crib decorated for Christmas.
Each player needs a small present he gives away.
The activity is usually embodied in some kind of Christmas celebration. The participants put their presents in the crib, then the celebration begins. At the end of the celebration, each participant may take one gift out of the crib.
Be sure to agree upon the types and sizes of the presents beforehand. May it be bought or does it have to be self-made, how expensive may it be, is there a minimum and maximum size, etc. No participant should have to walk away disappointed.