1Heat coals or gas grill for direct heat. Mix soy sauce, vinegar, oil and gingerroot in small shallow container. Place salmon in mixture, skin side up. Let stand 10 minutes.
2Cover and grill salmon, skin side down, 4 to 6 inches from medium heat 10 to 12 minutes, brushing once with marinade, until salmon flakes easily with fork.
3To serve, spread mayonnaise on crackers; top each with about 1 tablespoon flaked salmon, additional mayonnaise and green onion.