Heat oil in a large pot or dutch oven over medium heat. When hot, add the onion, carrot, celery, garlic, salt and pepper. Cook, stirring, until the veggies are just tender, about 5 minutes.
Add the broth, bay leaves, thyme, rosemary, oregano and cayenne. Stir to combine and increase heat so that it comes to a boil.
Once the soup is boiling, reduce the heat to a low simmer over low heat. Add the diced chicken (or turkey) and the uncooked wild rice. Cover and cook until the rice is tender, about 45 minutes.
After the rice is tender, uncover the soup and increase the heat so that is it simmering more vigorously, but not boiling (about medium heat).
In a small bowl, combine the dairy-free creamer and arrowroot flour and whisk until arrowroot has dissolved in the creamer.
While slowly stirring the soup, slowly pour in the creamer/arrowroot mixture until it is well combined in the soup. Continue to cook the soup, uncovered and simmering, stirring often until it has thickened, 5 to 10 more minutes.
Stir in the lemon juice, serve and enjoy!
