If you really want to use roundup, you can assure the land owner that unless a horse was to get out and go to a chemical warehouse and drink a barrel of roundup it will be ok. But if he is totally against using any chemicals, I would bush hog the plots and disc them several times until you get fluffy dirt. Keep disced every 2 or so weeks to keep weeds at a minimum.
Vinegar won't really kill the grass/weeds, it will likely dry them up some but the first rain they will come back. If you can a tarp ( like mentioned prior) would keep things at bay.