Process club soda, vinegar, tarragon, salt, tomatillos, avocados, cucumber, and jalapeño in a blender until smooth. Pour mixture into a bowl and let stand in a larger bowl filled with ice for 20 minutes (or cover and refrigerate for 1 hour).
Stir together shrimp, oil, and lime juice. Divide soup evenly among 6 shallow bowls. Place a mound of shrimp mixture in middle of soup in each bowl. Scatter tomatoes, bell pepper, and cilantro around shrimp mixture. Drizzle with additional oil, if desired.