A little research later, I figured out that the impeller had come loose from the pump motor. As is usually the case now, there is now way to fix this without replacing the entire assembly, which is very wasteful, as the motor is perfectly fine.
$90 and 15 minutes later, all is well in the laundry world again. This brings our spending on this machine to $160 over the last 15 months, but I can’t really complain, as it’s 13 years old. Hopefully with all these new parts, it’ll keep going for a long, long time!