They watched the TV show as children in the 1950’s. I read it in a science fiction magazine in the last 15 years. It was in either Asimov's, Fantasy and Science Fiction, or Analog. They saw only a limited number of shows and it had an “alien” host. They never met anyone else who saw the show. The sister spent her lifetime attempting to discover the history of the show and only occasionally discovered a small piece of evidence. In the end she never did discover anything about it. The implication was that it was not a TV show but really a transmission from an alien.
I found it in the September 1998 issue of Fantasy and Science Fiction. It is "Mr. Goober's Show" by Howard Waldrop. The intro to the story by the FSF editors say it was originally published in "Omni Online" shortly before it ceased publication. Thanks to everyone for looking.
Summary from this review:
An old man called Eldon recounts a mystical childhood experience that he and his sister Irene shared. While staying at their Aunt Joanie’s house, they encounter a mechanical television set that only plays one show: Mr. Goober. Eventually, their aunt gets rid of the television, but their curiosity about learning more about this Mr. Goober is too much, particularly for Irene. However, as a series of letters to Eldon reveal, Irene’s foray into the world of television as an adult slowly breaks her spirit down, while the discovery of what lies behind Mr. Goober may be too much for her to handle. So Eldon is forced to carry on the search in her memory, fruitlessly asking people over and over if they ever knew about this one show…
It is available on the Omni Magazine website.