Make the vinaigrette: Combine shallot, salt, and vinegar in a small bowl; let macerate 15 minutes. Add oil in a slow, steady stream, whisking until emulsified. Season with pepper.
Make the salad: Toss small tomatoes and squashes with 3 tablespoons vinaigrette. Prepare an ice-water bath. Bring a pot of water to a boil. Add beets, and cook until tender, about 20 minutes. Transfer to ice-water bath to stop the cooking. When cool enough to handle, peel, leaving stems intact.
Meanwhile, prepare another ice-water bath. Bring a second pot of water to a boil. Add snap peas, and cook until bright green and just tender, about 4 minutes. Using a slotted spoon, transfer to ice-water bath (keep water in pot simmering). Drain snap peas, and pat dry. Add potatoes to simmering water, and cook until just tender, about 10 minutes. Add potatoes to bowl with tomato mixture.
Preheat grill to medium. (If you are using a charcoal grill, coals are ready when you can hold your hand 5 inches above grill for just 5 to 6 seconds.) Grill potatoes, small tomatoes, and squashes, turning occasionally, until tender, 10 to 15 minutes. Toss remaining vegetables with 1/2 cup vinaigrette (reserve any remaining vinaigrette for another use). Arrange all vegetables on a platter. Season with salt and pepper. Garnish with basil.