So I hooked the trailer up to the van yesterday to test the brake controller.
The good news is that the brake controller works, and the trailer brakes work.
The bad news is that I am not getting 12V power from the van to the trailer. Last year I had a similar problem, but then it went away. At the beginning of the year, I didn't have 12V power. It wasn't mission critical, so I ignored it, but then in July I noticed that it had fixed itself, and it was good for the rest of the year.
So how, you ask, did I ascertain if the brake controller and brakes work if I don't have 12V going from the van to the trailer? well, I plugged the trailer into shore power, and that was enough to power the brakes and make sure that the controller/brakes work.
This year, having 12V from the van to the trailer IS critical, or the brakes won't function. So now I need to go at both the trailer AND the van with a voltage tester to figure out where the problem lies. Truth be told, I hope the problem is with the trailer and not the van, as the trailer will be much simpler to figure out what is what (and the wiring is more accessible).