Walker looks modern and athletic; historically it named a fuller. Fullers cleaned and thickened raw cloth by trampling it in water and lye—literally walking the cloth. In much of northern and western medieval England, that worker was called a walker; elsewhere you meet Fuller or Tucker for the same trade.
So Walker is an occupational surname from textile finishing, not from hiking. Scottish usage can also translate Gaelic “son of the fuller.” American Walkers may descend from English or Scots lines, or later adopters of the English spelling. The reunion-friendly gloss: cloth underfoot, not sneakers on a trail.