What was the first army that had a radio operator in combat units? By radio operator I mean someone who carried radio frequency communication equipment into combat and used it for communications purposes.
All major European armies in WWI had radios and radio operators from the very beginning. It was a new technology then, and some military commanders were reluctant to use it).
The previous major war involving a European power was the Russo-Japanese war of 1905; at this time radio was on the stage of experiments, and I have never heard of its use in that war (but see the "edit" in the end). I conclude that the first use of radio in real combat must have happened in the beginning of WWI, by all sides.
EDIT. I verified the comment of John Dahlman: radio was indeed used on ships of both Russian and Japanese navies in the war 1904-1905. Since 1904 Russians also used stationary radio stations on land in the same war. So perhaps the Russian-Japanese war was the first war where radio was used. (The difference between stationary and movable stations is not so sharp: a stationaty station can be disassembled and moved to another place. And movable stations were large and bulky at that time anyway.) The link that I found is in Russian but it has some very nice photos:
The first field radios evolved from the field telephones. In fact, the first field radios (such as the German Torn.Fu Series) could work in both modes as a radio and as a field telephone. However, while these units were portable, they required some additional set-up before use. That is, they couldn't instantly be used for communications.
The first "Walkie-Talkie" style radios were developed for, and employed, by the US Army in 1940, although similar units quickly followed in the other armies. These were self-contained radios with battery power, that didn't require additional set up. While they couldn't realistically be operated on the move by infantry, they did allow communication to be established very quickly when the operator stopped.