Heat vegetable oil in a large stockpot over medium heat; cook and stir onion and garlic in the hot oil until onion is softened and translucent, about 10 minutes. Add water, chicken base, potatoes, carrots, bok choy, and celery; bring to a boil. Reduce heat and simmer until vegetables are slightly tender, about 10 minutes. Add chicken; continue simmering until chicken is no longer pink in the center, about 10 minutes.
