Yes, vegetable soup with hamburger meat can be cooked in the slow cooker. If you want to cook hamburger soup in the Crock Pot:
To maintain the same level of flavor as the stovetop version, brown the meat and saute the onions, celery, peppers, and garlic in a skillet first — or skip this step for simplicity.
Dump all ingredients, except green beans, into the slow cooker and stir.
Cook for 7-8 hours on Low or 3.5-4 hours on High.
Add green beans when there are 2 hours left of cooking.