Your vinegar solution will never kill the grass or the weeds completely. This solution only desiccates the tops and does nothing to kill the roots. 

To remove the grass you have a couple options:

Dig it out or hire a sod remover if it is a large area which it sounds like it is. 

Use an herbicide but then you would have to keep your horse out of the area for some time. An effective herbicide will take 2 weeks to work.

You could also solarization the area. If the horse is in the area you would have to do this in sections so that the plastic doesn’t get trampled but this would fix your problem.Clear plastic has been found to be the most effective. It will take about 6 weeks. Slow but effective.

https://extension.umn.edu/planting-and-growing-guides/solarization-occultation

I would wait until the puddles are mostly dry. I think we are in for a few sunny days next week.