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