The idea of youkai and kami requiring belief to sustain themselves is highly reminiscent of Neil Gaiman's American Gods, where old gods exist, but their powers have waned due to lack of belief. As the novel was released in 2001, it's possible that later Touhou games were influenced by American Gods. As of writing this I am not currently aware whether ZUN has commented on the origins of this idea or not.
Gaiman's idea of supernatural phenomena requiring human belief is also linked to the real life spiritual tradition of chaos magic. Peter J. Carol's Liber Null (1987) is the ground zero for this movement. In a nutshell, chaos magic put human agency and the power of belief at the forefront, essentially stating that the way the world functions is dependent on belief. If someone believes strongly enough that magic works, it will work, and if sufficiently many people believe something, it will become true. These ideas most likely influenced Neil Gaiman when he wrote Sandman, as he would have been quite aware of them via his friend and fellow artist Alan Moore.
Human belief and attention influencing the spirit world is also related to the Western esoteric idea of egregores. Egregores are essentially collective ideas that have gained a life of their own. They both depend on human belief, but also influence those who come under their influence. This idea can be interpreted in purely social, cultural and psychological ways. Things like religion, ideologies, national identities and even fandoms certainly seem to have great influence on their members and a kind of life on their own. Indeed, the Touhou Project itself is at least an egregore in this sense. But egregores can also be interpreted in much more magical ways, as spiritual entities that collective belief and attention conjures out. Some even believe that egregores can then act out supernatural phenomena, empower their followers and even do things like create miracles.
To what extent these ideas filtered into Japan is an interesting question to which I have no answer. The Shin Megami Tensei series heavily uses real-life western esoteric concepts, and it's known that PC-98 Touhou games were influenced by SMT. Did the power of belief come from these games? Or did ZUN have his eyes on whatever is the Japanese equivalent of ”the magic scene”, his curiosity roused by the SMT games..?
Western mages in Eastern Wonderland
It should also be noted that characters depicted as being Western type or influenced magicians play a quite prominent role in the series. The face of Marisa Kirisame adorns the icons of every mainline Touhou game in the Windows era and she is as much a core part of the franchise as Reimu is.
The Grimoire of Marisa, which contains the ”sigils” for various Touhou characters, including herself, is an extremely fascinating bit of worldbuilding to say the least. However, rather than being an user of ceremonial magic associated with sigils, her practice of working with strange mushrooms is more in line with witchcraft and sympathetic magic. Another two places in the franchise which prominently feature various sigils are the Scarlet Weather Rhapsody and Touhou Hisoutensoku games, where each playable character has their own sigil that are used as graphical elements.
The much-forgotten Alice Margatroid is another Western magician. She's originaly from the PC-98 canon and in there she was the child of Shinki, creator of the demonic realm of Makai and of godly power. The Windows era canon makes no references to such, as her theme song might suggest she is from Bucharest, Romania. She controls various dolls via her magic. While it's quite the fantastic depiction, all sorts of idols and dolls have been used in real life magical traditions too. Often they are used in symphatetic magic as stand-ins for humans.
Patchouli Wisdom is another Western magician, though what she practices is actually a hybrid of Western and Eastern esoteric traditions. She is described as ”Eastern style Western magician”. Her magic uses the five Wu Xing phase changes (sometimes called five elements) and the sun and moon elements of the Japanese calendar. While the context of her magic is of course fantastical, manifesting as things such as fireballs and rain of mercury, the idea of trying to adapt Eastern ideas into Western magical context has happened also in real life magical traditions.
One Watcher, Triple Goddess and Myriad Fairies
The PC-98 canon featured many beings in line with Western sensibilities. One of these is Sariel, who was apparently rather ”heavily influenced” by the SMT games. Her name is a direct reference to one of the mythological grigori, Watchers, from which the term egregore is derived from. They were fallen angels in Jewish mythology that mated with humans and produced a destructive, monstrous race of giants. One of the Watchers was called Sariel, and was said to have taught humans the ”course of the moon” - the lunar calendar.
Another very interesting character that is rooted in Western esoteric tradition is Hecatia Lapislazuri. Her character is based on Hekate, an ancient Greek goddess of magic and boundaries. She has the honor of being the one Western deity to make an appearance in Touhou. Hekate is immensely popular within Western occult circles, as one would expect a goddess of magic to be. While Clownpiece is depicted as a ”hell fairy”, she is also called a lampad. Lampads are night nymps that are in service of Hekate.
Of course the fairies, featured very prominently in every single PC era Touhou game, can't be ignored either. While many think of them as purely fantasy creatures, they are actually of Western folkloric origin. In real life the fairies, or fae, were seen as a rather diverse group of entities. They were of ambiguous nature, sometimes hostile, sometimes benevolent to humans. Sightings of fae as well as disappearances caused by them are reported to this day, and striking similarities between UFO abductions and reports of being spirited away by the fae have been noted.
One can also find other scattered references to Western esoteric and occult concepts and symbols throughout the series. For me, a particularly memorable one is Sanae-sama's Grand Thaumaturgy spellcard. Pentagrams as symbols exist beyond Western magic, perhaps most notably in Taoist idea of Wu Xing, but thaumaturgy (ability to perform magical deeds or miracles) is most definitely a term from the Western tradition. It was quite striking to come face to face with that barrage of pentagrams the first time!
ZUN, champion of magic?
ZUN's knowledge and use of Western esoteric concepts shouldn't however be exaggerated. Much of it seems to derive from pop culture ideas about magic. It's very well in line with his ”Shanghai Alice” worldbuilding, a kind of fairy tale blending of Eastern and Western themes. He hasn't demonstrated a particularly deep understanding of actual Western magical traditions, yet I feel like something of substance has still filtered out into Gensokyo. These might simply be influence from the Shin Megami Tensei games, but ZUN does seem to have a talent for making interesting, inventive and accessible presentations of esoteric ideas.
The one thing that would speak of ZUN possessing some kind of deeper understanding of Western esotericism is the Grimoire of Marisa. Even if it was produced entirely unknowingly or intended for entertainment ZUN might have managed to produce something that somehow works when used within Western magical framework. At the very least, in the most banal, flattest sense, he has given an extraordinary tool for building a deeper connection with the world of Touhou. There are purely psychological interpretations of magic, and it most certainly works at the very least on that level. It's extraordinary for a creative like ZUN to hand out such keys to the fans, and one has to wonder if he in fact knew what he was doing and did it intentionally.
It wouldn't be surprising if ZUN, who calls himself the ”head priest of Hakurei shrine” and has said things such as ” I want to continue the activity of recognizing the impossibility of separation between the virtual and reality” would choose to act some kind of an emissary of magic. When you read the Hifuu Club stories, you start to get a sense that he pines for a more enchanted world, if such was not obvious from the way Touhou presents itself.
In the afterword to the Dateless Bar ”Old Adam” Hifuu Club CD he writes the following:
The true nature of chuunibyou is the heat of life.
It's an explosion of one's imaginative power.
t's a form of resistance against a chilled society, wielding both purity and creativity.
Lately, I've been thinking: how can I maintain my chuunibyou until the day I die?
If I can do that, I can enjoy sake for my whole life.
Would such a person not take the chance to spread magic to the world if given the chance?
Whatever the case may be, it's necessary for us to look at the Hifuu Club next to get the full picture of Western influence on Touhou.