Okay, though I am not very old, I will give you an IEM's history 101:
IEM is born in the development of In-Ear Monitoring System. During late 70s, various small companies (most have disappeared) started to develop small scare stage monitoring system for musician. One of the very first musician who used custom IEM on-stage is Stevie Wonder (on the early ~ mid 80s).
Around 1984, Steve Miller Band started to fund the development of full scare In-Ear Monitoring System to be used on tour. They tried and discarded several system until they adapted a system called 'Ear Monitor' on 1990 by Crystal-Taylor Sound (which is found by the same person who later established Future Sonics). Another musician, Todd Rundgren, also adapted the same system on 1986 when it was under development. Few years later as the technology got adapted by more musicians, a few other companies also developed similar systems (Circuits Maximus' C-MAX system, Bross Audio Designs' EarWorn system, Etymotic Research's CanalPhone system, Sensaphonics' ProPhonic system, etc).
By mid ~ late 90s, Sennheiser started to manufactured In-Ear Monitoring System as well. About the same time (1997), Shure enters the market (with earpiece made by Westone). It is not until the new Millennium that the earpeice (or universal IEM as we know today) that are once intended for In-Ear Monitoring System starts to become available to the general public, mostly due to the release and popularization of iPod at 2001. IEM isn't really much a new thing in comparison to mp3 player.
To answer your question, no one seems to know who made the first IEM, but Future Sonics is often credited as the first company that developed a successful In-Ear Monitoring System that resembles what we have today.
^I dug these info out when I was researching material for my IEM guide. It might not be totally correct, but I believe it is very close to the truth.