The correct use of the word “vicariously”

You could say that it is redundant, but on the other hand you could also say adding "vicariously" underscores the fact that you are experiencing the lives of people on the television, not just the sights and sounds you see on it.

Lots of words are redundant in the sense that you could figure out what the writer must mean even if he left that word out. But the word may still be useful by making clear the writer's intent, so that the reader doesn't have to figure it out or guess, or providing emphasis. Like, "He ran quickly down the street." Is it necessary to say "quickly"? Can one run slowly? But perhaps the writer wants to emphasize the speed. "She was severely injured when the saw cut off her arm." As opposed to being only slightly injured when her arm was cut off? Etc.

I suppose most of the time when you write a sentence like, "He lived vicariously through ...", the word "vicariously" is superfluous in the sense that living through something would always imply vicariousness. But one could imagine sentences where it would not necessarily be immediately obvious. Consider, "She lived vicariously through bad romance novels." You know exactly what I mean. But if I wrote, "She lived through bad romance novels", a reader could be forgiven for thinking that I meant that she survived the experience of reading the bad novels. Presumably I would mean "survived" here hyperbolically; I don't think a bad novel would literally threaten one's life. But people often talk of "surviving" an unpleasant experience. Maybe the meaning would become clear through the rest of the paragraph. But then the reader would have to backtrack and reinterpret the earlier sentence. Better to use words that explicitly say what you mean rather than forcing the reader to guess at your meaning.