Press out ginger juice (see below); whisk in soy sauce, vinegar, sugar, cornstarch, 1/4 teaspoon ground pepper, and 1 cup water; set sauce aside.
Season shrimp with salt and pepper. Heat 2 tablespoons oil in a large nonstick skillet over medium-high heat. Cook half the shrimp, turning once, until just opaque, 3 to 4 minutes; transfer to a plate. Add another tablespoon oil; repeat with remaining shrimp. Set shrimp aside.
Add remaining tablespoon oil to pan; cook bell peppers and white parts of scallions over medium-high heat, stirring, until crisp-tender, about 3 minutes.
Whisk reserved sauce; pour into pan, and bring to a boil. Cook, stirring, until thickened, 1 minute. Add shrimp and green parts of scallions; stir until coated with sauce, 30 seconds more. Serve immediately.
To make ginger juice, place finely grated ginger in a fine-mesh sieve set over a small bowl. With a spoon, press down firmly until juices are released. Discard the solids.