Sure! Here's a test case generator for the given problem "Shy Polygons":

```python
import random

def generate_polygon():
    n = random.randint(3, 14)  # Number of vertices (3 <= n < 15)
    vertices = []
    for _ in range(n):
        x = random.randint(0, 499)
        y = random.randint(0, 499)
        vertices.append((x, y))
    return n, vertices

def tcgen():
    test_cases = []
    
    while random.random() < 0.8:  # Generate multiple test cases
        L = round(random.uniform(0.1, 50.0), 4)  # Distance L (0.1 < L < 50.0)
        
        n1, polygon1 = generate_polygon()
        n2, polygon2 = generate_polygon()
        
        test_case = {
            "L": L,
            "polygon1": polygon1,
            "polygon2": polygon2
        }
        
        test_cases.append(test_case)
    
    return test_cases

# Example test case generation
if __name__ == "__main__":
    test_cases = tcgen()
    for case in test_cases:
        print(case["L"])
        print(case["polygon1"])
        print(case["polygon2"])
```

### Explanation:
1. **generate_polygon()**: This function generates a polygon with a random number of vertices (between 3 and 14 inclusive). Each vertex has x and y coordinates, both of which are between 0 and 499.

2. **tcgen()**: This function generates multiple test cases until a random stopping condition is met (controlled by `random.random() < 0.8`). For each test case:
   - A random distance `L` is generated between 0.1 and 50.0.
   - Two random polygons are generated using the `generate_polygon()` function.
   - The test case is stored in a dictionary and added to the list of test cases.

3. **Printing Test Cases**: The example at the end of the script shows how to print the generated test cases.

This generator will create diverse and random test cases within the given constraints, suitable for testing the "Shy Polygons" problem.