No yeast, no baking powder/soda
You can't make doughnuts without a raising agent--but you can make cannon balls.
What are cannon balls lol?
You could add some carbonated water (unflavored) to get some leavening. Or make a batter and fold in whipped egg whites.
Cannon balls are cannon balls! Don't waste your time trying to make doughnuts without leavening--you are setting yourself up for frustration. Get the yeast or baking powder and you'll have a better chance at making something edible!
Beer might work--if you were trying to develop a recipe and were willing to make numerous efforts. But for something edible the first try? I doubt it!