Rinse rice well and bring to a boil with 1 cup water and 1 teaspoon soy sauce in a 1- to 1 1/2-quart heavy saucepan, then reduce heat to very low and simmer, tightly covered, until water is absorbed, about 40 minutes. Remove from heat and let rice stand, covered, 10 minutes.
While rice is standing, stir together vinegar and remaining teaspoon soy sauce.
Transfer rice to a wide nonmetal bowl (preferably wood, ceramic, or glass) and sprinkle with vinegar mixture, tossing gently with a large spoon to combine. Cool rice, tossing occasionally, about 15 minutes.
Stir together wasabi and remaining teaspoon water to form a stiff paste. Let stand at least 15 minutes (to allow flavors to develop).
Place sushi mat on a work surface with slats running crosswise. Arrange 1 sheet nori, shiny side down, on mat, lining up a long edge of sheet with edge of mat nearest you. Using damp fingers, gently press half of rice (about 3/4 cup) onto nori in 1 layer, leaving a 1 3/4-inch border on side farthest from you.
Arrange half of cucumber in an even strip horizontally across rice, starting 1 inch from side nearest you. (You may need to cut pieces to fit from side to side.) Arrange half of carrot just above cucumber in same manner. Peel avocado half and cut lengthwise into thin slices, then arrange half of slices just above carrot in same manner. Repeat with radish sprouts, letting some sprout tops extend beyond edge.
Beginning with edge nearest you, lift mat up with your thumbs, holding filling in place with your fingers, and fold mat over filling so that upper and lower edges of rice meet, then squeeze gently but firmly along length of roll, tugging edge of mat farthest from you to tighten. (Nori border will still be flat on mat.) Open mat and roll log forward to seal with nori border. (Moisture from rice will seal roll.) Transfer roll, seam side down, to a cutting board. Make second log in same manner, then cut each log crosswise into 6 pieces with a wet thin-bladed knife. Serve with wasabi paste, soy sauce, and ginger.