With a regular fridge/freezer the steady-state relative humidity should be quite low because the cooling coils will condense out a lot of the water vapour, and drain it out a hole in the back to evaporate outside the enclosure. The cooling coil is always going to be the coldest surface within the enclosure, therefore that's where the water vapour would tend to condense.

Don't chest fridges have a similar mechanism? Perhaps the problem is things being too closely-packed, so the air can't circulate - so that when the lid is opened and water vapour let in from outside, it condenses on the food etc - but this isn't then removed by the mechanism above.