Heat the oil in a skillet over medium-high heat and cook the sausage, breaking it into small pieces, until browned, 6 to 8 minutes.
Add the onion, carrots and celery and cook until the onion softens, 4 minutes; stir in the garlic for 30 seconds.
Transfer the mixture to a 6-quart slow cooker and add the lentils, tomatoes, broth, oregano, salt and pepper.
Cover and cook on low until the lentils are tender but still hold their shape, 6 to 7 hours.
Stir in the spinach and vinegar and cover for 5 minutes until the leaves wilt.
Taste the broth after the sausage has seasoned it and add more salt only if needed.