Yes, a televised moon landing was predicted in one Golden Age story that I know of: "All Aboard for the Moon" (novel, 55000 words) by Harold M. Sherman in Amazing Stories, April 1947, available at the Internet Archive. Apparently never reprinted.
The following excerpt is part of Gil Benson's ("playboy; devil-with-the-women; and rich") speech just before taking off for the moon in
his atomic-powered spaceship, the Goodbye, World!:
"I'm also indebted to the General Electric Company of Schenectady for
permitting me to install a hitherto untried sending and receiving
radio set which beams radio waves of such high frequency that we are
confident they can penetrate both the Heavyside [sic] and Appleton
layers which surround the earth, at respective levels of sixty and two
hundred miles, so that we can keep in constant touch with this planet
during our travels and while on the moon.
"These new instruments, in conjunction with the television apparatus
we are carrying, will permit us to scan some of the moon's surface and
project back to earth the actual scenes as we are witnessing them. You
know, of course, that television waves travel in a straight line and
from the vantage point of the moon they can be beamed directly to
earth. In fact, could a television station be established on the moon,
we could then beam all television shows to the moon and relay them
back to earth on a straight line so that they would be receivable everywhere."
P.S. Here's an earlier (1940) but less clear-cut example. Not a science fiction story, it's a non-fiction feature titled "An Elementary Course In Astronautics" in Planet Comics #9, November 1940, available at Comic Book Plus. It doesn't mention moon landings per se, but predicts that spaceships of the future will maintain radio and televisual contact with earth. According to a statement in the previous issue, this issue went on sale September 10, 1940, a good five years before Project Diana, on January 10, 1946, demonstrated the feasibility of transmitting through the ionosphere by bouncing radar off the moon.
Quoting from the 4th and 5th panels on p. 30:
OBSERVERS ON EARTH WOULD CHECK ALL DETAILS WITH THE SHIP'S COMMUNICATION OFFICER.
A FORM OF ULTRA-SHORT WAVE RADIO AND TELEVISION WOULD KEEP THE ROCKET IN CONSTANT CONNECTION WITH THE EARTH.