I'm answering this as a Baseball fan of over 40 years ...
No team is to be credited with being first as far as walk off homeruns. All through baseball history games have occasionally been won with a last-at-bat home run, but the term "walk off" came into use after Kirk Gibson's amazing 9th inning home run in the first game of the 1988 World Series between the Dodgers and the Oakland A's.
The common understanding of "walk off" is where a batter on the home team hits a home run in their final at-bat that wins them the game. It implies that a batter, after hitting the home run, can almost casually walk around the bases to lengthen the celebration and then walk off the field. However, I have heard that originally the "walk off" term was coined by a pitcher, who the moment after giving up a no-doubt, game-ending homer, simply walked off the field without even turning his head to watch the flight of the ball over the fence.
Sportscasters, though, chose to use to term in an offensive sense compared to referring to the pitcher. Thus, baseball has another piece of terminology to describe this fairly common, game-ending moment.
baseballinsighter
Excellent answer to this question. The "walk-off" is a media term and is not an official, nor unofficial statistic in baseball. Raw.
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The all-time mark belongs to the Houston Astros, when they recorded five straight walk-off wins, in July, 1986.