In a large pot of boiling, salted water, cook the potatoes until fork-tender, about 7 minutes. Drain and transfer to a large serving bowl. Add 1-1/2 cups chickpeas.
Using a blender, puree the remaining chickpeas, garlic, vinegar and paprika with 2 tablespoons water. Pour in the olive oil in a slow, steady stream, processing until combined; season to taste with salt. Pour 1/4 cup dressing over the potato mixture and toss.
Preheat a grill or grill pan to high. Brush some remaining dressing on the mushrooms and zucchini, then season with salt and pepper. Grill the zucchini for 1 minute per side and transfer to a cutting board. Grill the mushrooms gill side up for 2 minutes; rotate 90 degrees and grill for 2 minutes more, then flip and grill until slightly charred, about 1 minute more.
Chop the zucchini and mushrooms into 1/2-inch cubes and add to the potato mixture. Add the basil and season with salt and pepper; toss to combine. Serve with remaining dressing on the side.