At a friend's house and she had a footstool assembled with what I
might call headless knobs, or screw-on rivets. They use a piece of
the threaded rod that are used for drawer knobs, in the middle, and on
each end was like a knob except that almost all of it was missing.
Just something like a rivet with a threaded hole in the middle, that
goes all the way through. The shaft is maybe 3/8" and the head is
maybe 1/2" or a little more and only a couple millimeters high, and
the whole thing is 1/2 long.

What do you call these things?

The current ones are brown, probably brown metallic.

Can I buy them at HD or a hardware store, or do I have to look on the
web?

This footstool matches some of the other furniture, all of which is in
v.good condition.