The recipe for chess pie includes cornmeal and vinegar-why?

Chess pie is characterized as a custard pie with cornmeal. Without the cornmeal it would cease to be a chess pie.

The cornmeal does thicken the filling. It also makes the texture coarser that other custard pies. Often, but not always, when I make them there is a chewier layer from some cornmeal that has separated. This texture difference is a lot of what makes chess pies delightful.

Chess pies often include acid to balance the sweetness a bit. This is not necessarily a dessert pie and tempering the sweetness is desireable. Given that it is a recipe popular in the south, recipes will often call for buttermilk. Vinegar is ok but the buttermilk versions are better, IMO.

Wikipedia has this to say about it:

Chess pie is closely related to vinegar pie, and the two terms are often used interchangeably. Vinegar pie generally adds somewhere between a teaspoonful and tablespoonful of vinegar to the above ingredients to reduce the sweetness. Some variations are called Jeff Davis or Jefferson Davis Pie, and Kentucky pie.

The vinegar is there to balance out the amount of sugar. If you leave it out, it may be sickeningly sweet. If you don't have vinegar, look up recipes for 'lemon chess pie', which use lemon juice instead.

As for the cornmeal ... it typically thickens the filling, to get it to the proper consistency. The typical replacement is a bit of flour, but it can leave an off (raw flour) taste if used in significant quantities.