Heat grill to medium-high. In a medium bowl, combine the vinegar, sugar, and ¼ teaspoon salt, stirring until the sugar and salt are dissolved.
Add the cucumber and onion; toss to combine. Let sit, tossing occasionally, for at least 15 minutes and up to 6 hours.
Meanwhile, in a food processor, pulse the salmon 3 or 4 times, just until coarsely chopped (still somewhat chunky).
Add the scallions, ½ teaspoon salt, and ¼ teaspoon pepper and pulse to combine. Form the mixture into ¾-inch-thick patties.
Oil the grill grate. Grill the patties, turning once (do not press or flatten), until opaque throughout, 2 to 4 minutes per side.
Place the burgers on the buns and top with the pickled cucumbers and onions.