Ben wrote:

That is, the bottom of the socket does not screw onto the lamp. All
hardware stores I've been to just have one-size of socket.

Is it a metric/imperial thing? I can't seem to find anything on
different socket mounting sizes, just tons of stuff on the different
bulb-mounts. I have no problem with a fitting a bulb in a socket, I
just can't find a socket that fits on my lamp.

You might try a lamp shop. The local one in my not-very-sophisticated
city was quite familiar with the thingie I needed for an old iron
lamp. Home Depot doesn't deal in specialties.