Do Buddhists use the Shambhawi mudra ?
What kind of tradition use it and for which purpose? (if possible with some references)
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This answer is only from the Theravada viewpoint. It's likely that you can find more diverse meditation techniques in Vajrayana.
Shambhavi Mudra is apparently an eyebrow center gazing gesture.
This source states:
Shambhavi Mudra is an eye-crossing technique which is used as a hack to activate the parasympathetic nervous system—which is cognate with the medulla oblongata and the ajna chakra (or the so-called third eye). A sign that the mudra is being practiced properly is the result of having pressure build in the sinus cavities.
While there is no Shambhavi Mudra in Theravada Buddhism, the closest thing to it is parimukham. Please read this answer for a detailed analysis on the term parimukham, with respect to the mindfulness of breathing (anapanasati) meditation, in the Pali suttas.
This is often translated as concentrating on the sensation of the breath at the tip of the nose in the mindfulness of breathing (anapanasati) meditation. That does not refer to the tip of the nose at the eyebrow center, but rather, it refers to the nostrils and area around the nose and mouth where the sensation of the breath can be felt.
But the sensation of the breath can also be felt at the diaphragm and abdomen. Please also see this answer.
The concept of chakras, and stimulating the nervous and endocrinal system is also not found in Theravada Buddhism.
On another note, there is however, something similar to Kechari Mudra in Buddhism, in this question. But again, the purpose is different. It's not to stimulate the nervous and endocrinal system.
The purpose of Buddhist meditation is to still the mind and use it to observe the processes of the mind and the mind-body connection, to understand the nature of the mind-body, and how it relates to the four noble truths, dependent origination and the three marks of existence. It's not meant for obtaining psychic abilities or supernatural powers - though that may be possible.