They load them just like they do anywhere else, someone goes into the house when a squad finishes and loads while the next squad shoots. When that squad is done they come out and go to the next trap and do the same. One loader can cover two traps and keep them loaded.
As for the trap machines, they're Pat traps. Pat turrets hold four cases (540 each) when full.
I don't know how many cases of targets are stored in the houses for each day's event(s). However, the supply is replentished in the evening. I noticed a crew near the shower house last year doing it after dark.