Identify Hot and Neutral Line on Vintage Clock
After more research, I realized that this question has been answered already -- I just didn't have the right terminology for it. When the prongs of a plug are the same size, they are "non-polarized" and don't have a hot and neutral.

I am looking to convert a non-polarized plug to a polarized one. According to this stack exchange thread it doesn't matter which way I wire it when going from non-polarized to polarized, so long as my gauge is the same or larger on the new plug so that it caries the old electrical load safely.