Here's the plain truth: nobody has a clear record of who invented the steam wand. We couldn't find a patent or a museum record that names a first milk frother, and we won't make one up.
What is documented is the machine it grew up on. In Milan in the early 1900s, Desiderio Pavoni built a café machine called the "Ideale" on Luigi Bezzera's system for brewing coffee one cup at a time. Steam wands are part of machines like these today, but when and where the first one appeared isn't documented.
In 1948 Achille Gaggia's lever machines, also from Milan, changed espresso itself. They pushed water through the coffee under high pressure, with no steam involved, and produced the layer of crema we expect today. The milk side of the story has fewer records. A frother at home does a small version of the café's work: heating milk and folding air into it.