I cleared out some work stuff, then took my desk apart and dusted everything for the first time in ages.
I then went out and looked at the air conditioning on the van again. This time I found the disconnected wire, and the compressor now runs. The pressure seems a little low (even with the new coolant I've added), but at least it's blowing cold air, rather than warm air now. I've zip tied the connection so it hopefully won't drop off again, but also so it'll be obvious if it does.