Prepare a grill for indirect heat. If using a charcoal grill, build the hot coals on one side only. If using a gas grill, heat one side only to medium-high heat.
Place a saucepot over direct heat. Add the vinegar, 1 cup water, sugar, salt, garlic and dill and bring to a boil while whisking to dissolve the sugar.
Add the zucchini to a large bowl and pour the pickling liquid over. Let stand until the zucchini are pickled but still firm, 5 to 10 minutes. Remove the zucchini from the pickling liquid and drain slightly in a colander set over a large bowl. Discard the garlic and dill.
To assemble the salad: Place the pickled zucchini onto a platter. Distribute the mozzarella and tomatoes evenly over top. Zest the lemon over top and squeeze the lemon juice all over. Drizzle with olive oil. Season with salt and pepper and serve at room temperature.
(Alternatively, you can boil the pickling liquid over medium-high heat on a stovetop.)