Most people associate qigong with Daoism, and rightly so. There is, however, also a qigong whose roots lie in Buddhism, more precisely in the combination of Chan (Zen) and esoteric Buddhism, in Japanese mikkyō. It's called Chanmi Qigong, and if you're interested in Reiki, you'll find a close relative here.
The reason starts with the script. Qi and ki are written with the same character, 氣. In China people speak of qigong, "working with qi," and in Japan of Reiki, "spiritual ki." The character is the same, but that doesn't make it the same energy. In Reiki the rei 靈 defines the kind: spiritual life energy, which you take in through the crown of your head after the initiation and which flows through you and out of your hands. Qigong, on the other hand, knows several kinds of energy, taken in through breath, food and exercises, for example.

What is Chanmi Qigong? 禅密気功
The name already tells you a lot. Chan 禅 is the Chinese reading of the character that is pronounced zen in Japanese, the meditation tradition centered on stillness and direct experience. Mi 密 means "secret" or "esoteric." It's the same character as in mikkyō 密教, the Japanese term for esoteric Buddhism, which Shingon 真言 belongs to.
So Chanmi Qigong brings together Zen meditation and esoteric Buddhist practice, with influences from Daoism and Confucian thought as well. Its basic principle is expressed in the sentence "We grasp the round and receive the One."
"Grasping the round" refers to the circular movement that runs through the whole practice. The body moves in spirals and circles, flowing and without a clear beginning or end. You could compare qi here to water, which finds its way around obstacles gently and persistently.

Usui and qigong 臼井氣功
In a Japanese qigong reference book, Mikao Usui is counted among the greatest qigong masters of the early 20th century, and Reiki is described there as a form of qigong, that is, as a method of working with ki. Outside Japan this is hardly known.
It makes sense once you look at the practice rather than the labels. In a Reiki treatment, the hands are placed, energy flows and the practitioner directs attention, nen, toward the person receiving. In qigong, the hands move, energy flows and the practitioner directs nen through the body. Both work with 氣 and both use the focus of the mind to guide it. In other words, the difference lies more in the form than in the essence. Chanmi Qigong also includes self-treatment, where the hands rest on the body and qi is guided by intention, which comes very close to Reiki.
The circle and the spiral 圓
The "One" in the sentence quoted above isn't a number but a state, the unity of qi and focus that arises when the mind becomes calm through the circular movement. In Chanmi Qigong this shows above all in the flowing, wave-like movements of the spine. The same circular principle is also used in martial arts, where soft and hard techniques rest on it, and something that looks soft from the outside can carry a lot of power.
Personally, I find the parallel to the spiral in the form of Cho Ku Rei interesting, the first Reiki symbol, whose line winds inward toward a center. More on how qigong and Reiki relate to each other is in the article on qigong and Reiki.
Qigong and Reiki in everyday practice 日々
Qigong and Reiki complement each other well. Qigong brings the body into movement and loosens what has become stuck, while Reiki leads into stillness and gathers the attention. My own day starts with Reiki, because I lay my hands on myself every morning right after waking up, to wake up and start the day relaxed and motivated. Movement practice fits in well before or after.
Combining the two isn't a modern experiment. The roots of both lie in esoteric Buddhism, Shugendō, shamanic Daoism and Shintō, and the fact that they're often seen as separate today has more to do with how they spread than with the practice itself. Do you already combine movement and stillness in your own practice?

Discover the shared path
Reiki and Qigong share the same root. If you're curious, have a look at how both come together in Shingon Reiki.
Your Path in Shingon Reiki Qigong and Reiki