No. You don’t need to believe in anything. In fact, acupuncture has been scientifically proven to work very well on horses and dogs and has been approved as a veterinary medicine. The horses and dogs don’t follow a belief system and it works for them, and it’s the same for us all.
Acupuncture developed first in ancient China which had three main philosophies at the time: Daoism, Confusionism, and Buddhism. Traditional Chinese Medicine has gone through many political upheavals which changed the way the medicine was practiced or even whether it was practiced at all. Currently, a lot of the TCM literature has references to Daoism, using the language of Yin and Yang (balancing opposite energies) and the five elements. Many of the traditional medicines around the world are based on this concept of balancing the elements and energies of the body.
So if we can see that most traditional medicines share this common thread, and they have worked for centuries, then it doesn’t matter whether the language used in TCM sounds Daoist, and we are not Daoist. I am not a Daoist either. The Ayurvedic medicine of India uses five elements, Tibetan medicine uses three energy types based on 5 elements, and ancient Greek medicine which gave birth to modern Western medicine, had the law of the four humors, another elemental energy system. This law was used both in early Western psychology as well as being the philosophical root of modern Western herbal medicine before the medieval era which saw the burning of the “witches” (largely herbalists) and their herbal books.