Are there any Jewish sources which discuss the existence of auras or energy fields? Is it permissible to learn about them and be involved with them for healing purposes?
There are different groups/individuals who discuss these ideas. Certainly, this use of the term "energy" has nothing to do with the scientific definition of the word. This "energy" idea has pagan roots and is known by many different names- In Hinduism the terms prana and chakra are used. In Chinese systems the terms are Chi or Qi. And there are many other variants. Most major poskim (rabbis who are expert in practical Jewish law) have expressed concerns with these ideas/methods and some have forbidden to be involved with them.
אין היתר להשתמש ברפואות אלטרנטיביות שמקורם בדעות עובדי עבודה זרה ויסודם בפעולת כחות לא טבעיים ־ אלא אם כין אפשר להוכיח שהוא עובד בדרך ״הטבע״ בדרכי הבירור, ולא סגי בכך שהעוסקים ברפואות אלו מאמינים שהוא ״טבע״. ויש לחוש בזה משום קוסם קסמים מעונן ומנחש, או משום דרכי האמורי. וקל וחומר אם לא השמיטו מכלל הלימודים את חלק הע״ז שבדבר, שיש בזה איסורי ע״ז
Rabbi Moshe Tendler is well known for being against unproven alternative medicine and probably forbids "energy" medicine as well.
The most pertinent aspect of Jewish law here is the prohibition against superstitious practices. That is codified in Shulchan Aruch, OC:301. The basic idea is that unproven (or perhaps dis-proven) remedies which are not based on science are forbidden. There are also may be other prohibitions involved, including divination and idolatry. (I need not mention the general obligation to guard ones health, and seek out medical intention in case of illness.)
Of course, there are subtleties involved and this answer is not meant to substitute for a competent rabbi's decision. Be sure to ask a rabbi who knows the subject well, as many people are simply unaware of the pagan roots of some alternative medicine and may be unaware of the medical literature that has proven them ineffective.
In kabbalistic and other rabbinical literature, the terms "nefesh" (and sometimes "ruach"), understood as the lower souls, or levels of the soul, are often described as consisting of a "life energy" pulsating through the body, which is similar to the concept of "qi" found in Chinese medicine, and similar concepts found in many cultures worldwide.
There are also more general references to a "vital spirit" within chassidic literature. For example, here are a couple quotes from Rebbe Nachman of Breslov, which describe the nature of this "vital spirit."
"From the very beginning of life there is a vital spirit in the heart which causes it to contract and beat. This beat affects all the blood and fluids in the body. It beats in all the limbs of the body, and by constantly beating and churning them, it keeps them from stagnating and degenerating, just as the wind blows over the sea and beats and stirs the waters so that they do not stagnate. Most importantly, this beat pulsates in the arms and hands, which are constantly active and therefore need this beating, churning and cleansing more than anywhere else in the body. This is why the doctor can discover everything about a patient's condition by laying his hand on the patient's wrist and taking his pulse. This is because the arms are the main place where it is possible to feel the action of the heartbeat, which is responsible for the life of the whole body" (Likutey Moharan I, 56:9).
"The depression which descends on a person when he has to struggle excessively for a living is the `filth of the serpent.' All the limbs of the body become heavy, and the vital spirit which pulsates in the body -- the very basis of life -- is weakened. The weaker it becomes, the heavier the limbs become, and they in turn weigh down the spirit even more. This vicious cycle can actually bring a person to the point of death." (Likutei Eitzot, Money and Livelihood).
Some rabbis have disapproved of learning about or participating in energy-based practices. Rabbi Yitzchak Ginsburg, a well-known Israeli kabbalist, posits that alternative healing practices can harness energy "from impure sources."
However, such negative appraisals are not universally held. Rabbi Yerachmiel Fried, for example, seems to approve of practicing Tai Chi (including sensing the flow of life energy stimulated by the exercise) in this article:
Rabbi Zamir Cohen in his book the discusses Auras as a scientific fact. Which would assume that halacha permits it. The following is a footnote from one of his articles https://www.hidabroot.com/article/174198/Conflicting-Forces-vs-Harmonious-Duality-Rabbi-Zamir-Cohen where he quotes his own book. and discusses auras.
Working with peoples energy fields is accepted in halacha. Although I don't feel comfortable quoting the name of the rabbis without their permission I was told by two separate rabbis that working with energy is completely permitted as long as there are no rituals involved. As always for practical halacha ask your own rabbi.
By "aura" I understand you to mean a field of light emanating from people, varying in colour and intensity according to the physical, emotional or mental state of the person. I don't have a written source, but I heard the following anecdote from an internationally popular Rabbinic speaker:
He was speaking about head-covering and why married women cover their hair but men do not, and he said (in confidence, so I am not mentioning his name) that he had a friend who could see people's auras. This friend could tell whether his wife was covering her hair by the visibility of her aura. She would stand behind a door and put her hand around it and when her hair was covered he saw no aura around her hand.
Cute, but not very scientific... :-)