It's difficult to trace the beginning of this idea in engineering. In some regards it's like the antennas that have been dangled from balloons since at least the 1930s. But that video above is using air, not even hot air, and certainly not helium. It's height and wave length rely on air pressure, not buoyancy. Knowing that much, here are some predecessors that come to mind.
In 1956 NASA began work on an inflatable sphere they called Echo. It's purpose was to passively reflect signals back to earth for communications. The first models were literally just balloons covered in reflective foil. The real Echo I was a 100-foot wide Mylar and aluminum balloon. It's stopper was basically a bag of water. They failed at a small scale test launch in October 1958, but they pressed on. Echo I launched August 1960. While that was pretty nifty. That video above shows both an antenna and tower in one conical balloon. Echo was basically a parabolic dish, though air pressure is what allowed it to function. It's reflecting, not broadcasting. It's not quite the same thing.
I did also find a record of an inflatable loop antenna in a NASA white paper from 1967 about ELF (Extremely Low Frequency) radio experiments on OGO 4. Details are as follows:
"This experiment consisted of six VLF radio receivers that studied natural and man-made VLF noise occurrences at orbital altitudes. The receiver systems consisted of an inflatable 2.9-m loop antenna, a preamplifier stage at the end of a long boom, and a receiver electronics package in the main body of the satellite. Three step-frequency receivers covering frequency ranges from 0.2 to 1.6, 1.6 to 12.5, and 12.5 to 100 kHz each observed a complete spectrum of 256 signal strength values once every 4.6, 18.4, or 73.7 s..."That is definitely an inflatable antenna, but it's receiving, not broadcasting. I'm nit-picking as obviously it could have been built to do both but it appears that it was not used in that fashion. That brings us to GATR and LTA. GATR is a bit bigger and less-portable than the LTA hardware. It packs into a three-case, system and in the end you have a very portable satellite dish. LTA Projects have a few different models. They too advertise portability, but are also marketing themselves for emergency and temporary use. But their units are portable enough to mount on a vehicle, but rugged enough to stand in 35-mph winds. At 35 mph I can only imagine that satellite-ball rolling across a parking lot...