The freezer draft fan was rattling.
After some googling, I turned the refrigerator off, emptied the freezer, and used a hair dryer to melt the ice that had accumulated around the fan.
After melting the ice, and turning the fridge back on, the fan was still rattling, but when fiddling around, I pushed the fan shroud down so that it was flush with the freezer floor, and the rattling stopped.
It turned out that ice that formed on the freezer floor had pushed the fan housing upwards a couple of millimeters, so that the fan was rubbing against the shroud.
It also turns out that my freezer is overstuffed. You should only fill your freezer half-full to allow enough space for the cold air to circulate.
The coils behind the back panel of the freezer may also have a worn out defroster bulb and thermosistor that can also allow ice to accumulate. Both should be replaced at the same time.
Not overstuffing my freezer may also help keep the door sealed, which will help with reducing the workload on the defroster bulb.
Nice work!
Somewhat related, if you ever have the opposite problem (not enough in your freezer to make it worth running), you can get sheets of foam insulation and cut them to fit. I filled up the unused volume in my deep freeze this way and now use it for what I need without worry.

