Shamanism similar to the ones seen in indigenous Siberian Cultures are rare in Japan, but there are few rituals of summoning a spirit and letting it possess people to ask practical questions or to interact with in general. One of which is Kitsune-odori [狐踊り] (Itachiyose [イタチ寄せ] in some case) (not to be confused with the choreographical Himeshima-bon'odori [姫島盆踊り] of Hime Island [姫島], Ōita Prefecture) seen in Northeastern and Eastern Japan.
In Hinoemata Village [檜枝岐村] (Minami-aizu Dist. [南会津郡], Fukushima Prefecture), Itachiyose is conducted in Hiuchigatake Shrine [燧ヶ岳神社] in near Mt. Hiuchigatake [燧ヶ岳] (below). First, the medium would sit in the middle of the shrine with a rope in their hands as 5 to 10 people began revolving around them chanting:
「だいけんにっそん日の神、だいけんがっそん月の神、しんとうかじ。」
"Daiken-nisson-hinokami, daiken-gasson-tsukinokami, shintōkaji"
Though I have no idea what "shintōkaji" means, "daiken-nisson-hinokami" and "daiken-gasson-tsukinokami" are probably referring to Sūrya/Āditya (or Nitten [日天]) and Candra (or Getten [月天]) based on how the participants would seal a symbol (like this one here which can also be seen in Mikkyō Buddhist) when the summoning is completed after the medium's body violently trembles as they jump up without using their hands.
While in Ubajima [祖母島] (Shibukawa City [渋川市], Gunma Prefecture), there were instances of Kitsune-odori up until Taishō Period where children would play rock, scissors, papers and the one who lost would be blind folded as the other kids began chanting:
「あさ山は山、羽黒の権現、ならびのとうかの大明神。」
"Asayama (Mt. Asa)-hayama (Mt. Hayama), haguro-no-gongen (the deity of Mt. Haguro (Yamagata Prefecture)), narabi-no-tōka-no-daimyōjin (and the Inari deity)"
When the blind folded child is possessed, the child would quickly jump up and began playfully chasing other children with an abnormal speed and if a child manages to catch the possessed one, the spirit would then possess the child who caught.
Other Kitsune-odori worth noting is from Tama Ward [多摩区] (Kawasaki City [川崎市], Kanagawa Prefecture). Until the beginning of Meiji Period, a child would sit in the middle of other children who then began repeating:
「亀の甲山のおろくさん、お花遊びにいらっしゃい。」
"Oroku-san of Kame-no-koyama, come play with us."
Then after the possession, the children would ask: "Kitsune-san, Kitsune-san, dokokara-kimashita?"/"Mr. Fox, Mr. Fox, where did you come from?" And after the possessed child answers, the other children thanks the vulpine spirit and asks the entity to sing a song to them. After a while, the vulpine spirit tells the children goodbye and leaves the body as the previously possessed child falls over, but after finger writing the character 水 (water) on his/her back, the child wakes up.
From the 3 examples above, one can notice that Kitsune-odori was practiced casually by commoners without religious intentions.
Unfortunately, there are no recent record of such rites being practiced these days.
Source: "Dōsojin-to-jizō" [道祖神と地蔵] (1994) by Tatehiko Ōshima [大島 建彦] (1932-present)