To expand on @plannapus' answer:
Chinese folk religion or Taoism doesn't worship Sun Wu Kong (孫悟空
) per se, but rather a monkey deity usually referred to as the Great Sage (大聖
) or Heaven Rivaling Great Sage (齊天大聖
). Historically, this has been particularly prevalent in Fukien, but stories similar to Sun Wu Kong's backstory can be found in several regions. This predates the c. 1592 Journey to the West, but the popularity of that novel furnished the traditional monkey deity with a standardised (sort of) backstory and the name Sun Wu Kong.
Roots of the modern monkey deity worship certainly predates the late 16th century fictional work, however. From about the late Yuan dynasty, elements of traditional folklore were incorporated into fictional works. A play by the Mongolian playwrite Yang Jing-xian, confusingly also titled Journey to the West, contains an autobiographic line by a certain "Itinerant Monk Sun" (recycled into the much later novel as a common title for Sun Wu Kong):
《元·楊景賢·西遊記·第三本》小聖弟兄、姊妹五人,大姊驪山老母,二妹巫枝祗聖母,大兄齊天大聖,小聖通天大聖,三弟耍耍三郎
I have five brothers and sisters. Big sister is the Old Lady of Mt Li; little sister is the Wu Zhiqi Sage; big brother is the Heaven Rivaling Great Sage; I am the Heaven Reaching Great Sage; and little brother is Shua Shua.
Note that here, Xuanzhong's monkey companion has a name (Heaven Reaching Great Sage) that's different form its usual modern version (Heaven Rivaling Great Sage). Both names do however appear for the monkey brothers in contemporary works such as the play 二郎神鎖齊天大聖
or the novella 陳從善梅嶺失渾家
. They are also found in an archaeological site on Baoshan mountain in Shunchang, Fukien, known as the Twin Sages Tomb. it's located near the Southern Heaven Gate, a temple first built in 1391 and rebuilt in 1548, which some scholars argue are the upper and lower bounds for the tomb's age.

(順昌寶山雙聖墓 - Twin Sages Tomb, at Baoshan, Fukien)
The inscriptions are difficult to read from the photo, but the left is the spirit tablet for a certain Heaven Reaching Great Saint (通天大聖
), while the right is for the more familiar Heaven Rivaling Great Saint. The positioning (left is senior) and style of the steles matches the play's version that the Heaven Reaching Great Saint is the younger brother.
On this basis it is thought that the Journey's character of Sun Wukong - and the monkey deity - is in fact an amalgamation of both Great Sages and their sister, Wu Zhiqi. That is in fact a considerably more ancient name, tracing back to at least ninth century. During the Tang Dynasty, Li Gongzu, a fantasy novelist who penned the famous Governor of Nanke, wrote that:
《唐·李公佐·古岳讀經》 禹理水,三至桐柏山,驚風走雷,石號木鳴,五伯擁川,天老肅兵,不能興。禹怒,召集百靈,搜命夔龍,桐柏千君長稽首請命,禹因囚鴻蒙氏、章商氏、兜盧氏、犁婁氏,乃獲淮渦水神,名無支祁,善應對言語,辨江淮之淺深、原隰之遠近。形若猿猴,縮鼻高額,青軀白首,金目雪牙,頸伸百尺,力逾九象,搏擊騰踔疾奔,輕利倏忽,聞視不可久,禹授之章律,不能制,授之鳥木由,不能制,授之庚辰,能制。鴟脾桓木魅水靈,山妖石怪,奔號聚繞,以數千載,庚辰以戰逐去,頸鎖大索,鼻穿金鈴,徙淮陰之龜山之足下
While trying to control the flood, Yu came to Tongbai thrice ... [there he] captured the river god of Huai. It is named Wu Zichi and shaped like a money, with a sunken forehead and tall nose, a green body and white face, golden eyes and white teeth ... With a chain tied around his neck, [the monkey] was imprisoned under the Turtle Mountain of Huai.
While it might only be his own fictional invention, the story was collected into 戎幕閒談
, a compilation of strange tales by the Tang writer Wei Xuan (韋絢
), as the explanation for a supposed incident he heard:
時有漁人,夜釣於龜山之下,其釣因物所制,不複出。漁者健水,疾沉於下五十丈,見大鐵鎖,盤繞山足,尋不知極,遂告湯。湯命漁人及能水者數十,獲其獲,力莫能制,加以牛五十餘頭,鎖乃振動,稍稍就岸。時無風濤,驚浪翻湧,觀者大駭,鎖之末,見一獸,狀有如猿,白首長口,雪牙金爪,闖然上岸。高五丈許,蹲踞之狀若猿猴,但兩目不能開,兀若昏昧,目鼻水流如泉,涎沫腥穢,人不可近。久乃引頸伸欠,雙目忽開,光彩若電,顧視人焉,欲發狂怒,觀者奔走,獸亦徐徐引鎖,拽牛入水去,竟不複出
A fisherman was fishing by the Turtle Mountain at night. His hook was caught on something and would not budge. Knowing how to swim, he dived in 50 yards, and saw a seemingly endless, large iron chain surrounding the foot of the mountain. He reported what he saw to the governor, who sent dozens of swimmers and more than 50 bulls to pull the chain out. There was no wind at the time, but to the shock of bystanders huge waves formed on the river. At the end of the chain, there was a beast, shaped like a monkey/ape, white faced and long moused, white teeth and golden eyes. He stumbled onto the shore, some five yards tall, moving like a monkey but with his eyes closed ... [after a while] his eyes suddenly opened, looking around like lightening, and looked around like he's about to go on a rampage. Everyone started running off, and he slowly walked back into the water, dragging the bulls with him, and vanished.
Furthermore, the Tang scholar Li Zhao records in his Addendum to Tang Histories (唐國史補
) that:
楚州有漁人,忽于准中釣得古鐵鎖,挽之不絕,以告官。刺史李陽大集人力引之。鎖窮,有青獼猴躍出水,复沒而逝。後有驗《山海經》云:水獸好為害,禹鎖于軍山之下,其名曰無支奇
A fisherman from Chu found an old iron chain in the Huai River. He pulled on it but it went on endlessly, so he reported it to the government. The governor gathered many men to pull. At the end of the chain, a green monkey lept out of the water, then vanished. Consulting the Classics of Mountains and Seas found a passage saying: A water beast was being a pest, and Yu locked it under the Jun Mountain. Its name is Wu Zichi.
The Classics of Mountains and Seas cited in the last passage is an ancient bestiary that dates back as far as the fourth century B.C., but the extant versions do not contain this supposed quote. It is quite possible that this part is a fabrication - citing old documents lend credence to the story. Nonetheless, the numerous references to the story suggests that, while the specifics might be inventions of writers, the idea of a monkey river god was an established concept in Chinese folklore.
Moreover, there are obvious parallels with the Sun Wukong character from the Journey to the West. The imprisonment under a mountain is particularly notable, and the physical description of the monkey Wu Zichi's golden eyes is an obvious precursor to Sun Wukong's fabled true seeing Flaming Eyes and Golden Iris (火眼金睛
). These similarities are why Wu Zichi is widely considered the origin of Chinese folk religion's monkey deity, the Great Sage.
An alternative school of thought argues that the Great Sage was inspired by the Hindu monkey god Hanuman. Whether a native Wu Zichi absorbed influences of Hanuman, or whether Wu Zichi itself was inspired by Hanuman, is a matter of some scholarly debate. Regardless of the true sequence of events, a tradition of a monkey deity most likely found itself transplanted from India into China. An (originally) Tang era Hindu-Buddhist Kaiyuan Temple at Zayton contains a stone mural widely considered to depict Sun Wu Kong.

(泉州開元寺仁壽塔 - Renshou Pagoda, rebuilt in 1237, at the Kaiyuan Temple, Fukien.)
This "Itinerant Monk Sun (Monkey)", as the mural is often called, predates Journey to the West by centuries.
Since the Ming Dynasty, Fukien has been a primary source of Chinese emigration and colonialisation. Unsurprisingly, veneration of the monkey god was transplanted by Hokkien settlers to Taiwan and South East Asia by Hokkien settlers.