This is the first thing that comes to my mind.
You start with a burrow with just the trainee.
Put a meeting area just under a hatch. This will cause the trainee to go there whenever idle.
Put a pressure plate attached to that hatch in the meeting area, so that stepping on the plate opens the hatch. This will mean that whenever the trainee is idle, the hatch opens, dropping whatever items are on top.
Next, you set up a loading mechanism.
Put a stockpile on the same tile as the plate, or next to it if that isn't possible. Set a minecart to load from the stockpile. Set it to be guided every day. Set it to dump on top of the hatch. (build ramps, stops and track as appropriate.) Some refinement may be necessary to give the hatch enough time to close before the cart is dumped.
When the dwarf finishes loading, they become idle and receive their daily allotment of socks. Once the socks are delivered, the dwarf then ceases to be idle and reloads the sock machine.
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This can be further refined by having a multi z-level path of hatches & plates so that when the dwarf becomes idle after dumping the cart, in order to get to the meeting area, they are forced to walk across the pressure plates opening each hatch in turn.
I haven't yet tried the dwarven shotgun, so there may be refinements/simplifications possible using that idea. And if you can create a minecart light enough to not injure the dwarf, that makes the whole plate/hatch/stockpile thing superfluous.
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I guess now we know what happens to the socks that disappear when you do laundry.