Can I broadcast a fall foodplot, in a beanfield, or is it too late? And if so can I spread it after the field is picked?

It all depends on where you're located and what you're planting.  I've broadcast oats and cereal rye into standing or cut beans into the middle of September and still had good growth.  You're probably a little too late for brassicas, which I like to plant in late July to mid August here in Illinois.