I need to make this recipe without eggs or dairy (butter is okay). I was going to sub. a vinegar and almond milk mixture. Will that work? I also n...

Kayla, I think that a vinegar + almond milk mixture would likely work but are you omitting the eggs as well? This recipe is a little forgiving as it's baked in a skillet, which gives and holds its shape, but since it does lack flour it makes for a very crumbly bread. The gluten in the flour is a protein that creates elasticity in dough and structure in breads and baked goods. Are you trying to make the bread completely vegan?

A vinegar+milk sub will be fine in this for buttermilk, you need the acid to activate the baking soda but the milk proteins aren't really doing much else in this recipe. The bigger problem is going to be the eggs. Those are the main structure to this bread.
You noticed the lack of flour. Usually what that does in corn bread is make it more "bread"-like, and helps it hold together. The gluten that wheat flour has helps those cornbreads with cohesion. This recipe compensates some for that with the additional eggs. Without them it would be a crumbly mess.

To make this without eggs, you'll want to use a "structural" replacement, like flax and water or silken tofu. (Other replacements like Baking Soda+Vinegar or Banana replicate the fluffiness or moistness of eggs, neither of which are really what you need here.) Adding 1/2 to 3/4 of a cup of flour will help as well. I would add it a bit at a time until you have a nice, thick batter.