Vinegar and lemon juice have pretty much the same pH (degree of acidity). I've substituted lemon juice for vinegar in any number of recipes, and it always works.

And the recipe, as given above, calls for baking powder (which contains its own source of acid), not baking soda (which needs an acid ingredient to react with). Furthermore, the brown sugar and the cocoa powder (assuming it's "regular" cocoa, not "Dutch" - treated with alkali) are both pretty acidic. Because of these factors, I'm not sure what the vinegar is even supposed to do here.