Contrary to what many think, video calling had been invented way earlier. Although it wasn't portable, and much like a sophisticated big TV set.
The man who deserves the attention here is Dr. Georg Schubert. He had been working on television transmission and reception, among other things, in a company called Fernseh in Germany.
The earliest patent talking about transmission of animated images on a live basis, just after they have happened is US2079880, filed in 1932.
It is also interesting to note that AT&T showcased a 'PicturePhone' allowing speakers to see regularly updated snapshots of each other at a fair in 1956, one year after the death of Dr Schubert. Following this, many attempts were made to have the video calling feature we have now.
Few interesting articles to read about this: here and here