Every Japanese is Shinto when he is born, a Christian when he marries and a Buddhist when he dies.
Most weddings in anime are Western-style (i.e., Christian or Christian-influenced) ceremonies, with white "princess"-style gowns, tuxedos and rings.
In Japan, the civil or religious ceremony of a wedding has no legal status — the actual marriage is enacted by filing the proper paperwork with the government and is all that is needed; anything else is simply an occasion for a party. Given that most Japanese do little more than give lip service to religion for most of their lives, it should not come as a surprise that they do not feel any particular need to have Shinto or Buddhist ceremonies. In fact, traditional wedding ceremonies are very long, very complicated, extremely uncomfortable for the bride, and (most importantly to a young couple and their families) expensive, even compared to the most excessive American weddings.
Western-style weddings, like many things Western/American, are considered "cool", and in this case romantic and elegant as well. Combine that with the substantially lower cost of a Western-style wedding (yes, Eastern-style weddings are even more expensive), and it's not surprising that it's a common enough choice that it's not even considered outré or "foreign" any more.
Correspondingly, because of their "cool"/"romantic" factor, Western weddings are the way to go for most animé characters. And they're just more fun for the artist to draw, too.
Japanese brides frequently have multiplewedding dresses, and change between them at various points on their wedding day. Changing wedding dresses is known as oiro-naoshi, and having many expensive wedding dresses to change between is a way in which Japanese brides and their families can conspicuously flaunt their wealth. The oiro-naoshi is also a way to let the bride have a Shinto wedding (in traditional white kimono) followed by a Western-style reception.
The grooms, for whatever reasons, generally favor a style of tux not seen unironically in the Western world since about 1980 (see above picture). Ruffled shirts, tailed jackets, pastels and white-on-white seem to the order of the day.
This isn't a Japanese Media Trope so much as a Japanese cultural trope (although the tuxes may be informed by media depictions). Other East Asian countries also prefer a western wedding instead of their respective tradition.
The early Magical Girl show Wedding Peach features heroines who fight in Western wedding gowns. The series is centered around wedding themes, and includes lots of information about Japanese weddings. The Magical Girls also perform the oiro-naoshi to change into more traditional magical girl costumes halfway through most battles.
Although it has not yet been animated, Love Hina ends with Keitaro and Naru's Western-style wedding. Keitaro also fantasizes western-style weddings a couple of times — in the first episode, with the unknown girl of the Childhood Marriage Promise, later with Naru.
There's also a brief scene in volume 8 where Keitarou feels like his life slipping away from him, and in particular, imagines Mutsumi (as the groom) and Naru (as the bride) getting married and leaving him in the dust...
As a counterexample, the last episode of the second Sakura TaisenOVA series depicts a formal Shinto wedding as held by very traditional Japanese family.
In the chaotic and completely ruined wedding that caps off Ranma ½, (well, in the manga, anyway) Ranma's in a tux and Akane in a Western wedding dress. Additionally, at the beginning of the arc, Soun, Genma, and Nodoka dressed Akane in a white kimono (after knocking her out) and proposed they just do it right then.
Much earlier, during Mousse's re-introductory arc, there is a confused moment where everyone believes Akane has been turned into a duck and Ranma is thus forced to marry "her" (long story). The result is a hastily performed mishmash Shinto ceremony.
An anime-only episode has Happosai apparently dying, whereupon Soun and Genma decide that Ranma and Akane must marry as soon as possible.
Also, Shampoo and Ranma have been depicted as having a Western wedding ceremony in the "Curse of the Reversal Jewel" story, and both a Shinto one and a Western one in the anime-exclusive "Shampoo's Red Thread of Dread". In the latter case, the first "wedding" occurs in Akane's imagination, while in reality Shampoo receives a Western-style dress, with she and the tuxedo-clad Ranma actually going to a church to conduct the ceremony.
For some reason, when the main characters of Maison Ikkoku get married at the end of the series they decide on a traditional Shinto wedding despite being dirt poor.
This is probably because Rumiko Takahashi often uses her works to try to preserve knowledge of Japanese culture and traditions in the face of increasing Western influences.
Well, they did get the money from Grandma Godai, who basically told them they should have a traditional wedding.
If memory serves, they also decided on a traditional wedding because Kyoko's marriage to her first husband had been Western. Kind of a way of saying, 'My husband died and nothing will replace him in my heart, but I can love someone else without it being an insult to his memory.'
The Godai's may be a bit traditional, one of Godai's (many) daydreams has him in a shotgun! Shinto wedding to Kozue, while Kyoko and Mitaka are seen in western dress (Ironically, Mitaka's actual wedding was almost certainly Shinto).
Averted in a flashback sequence in Hell Girl, which shows that Hajime and Ayumi married in traditional Japanese style.
An episode of Kodomo no Omocha has an engaged couple arguing whether the bride should wear a traditional kimono or a Western wedding dress.
The Tokyo Mew Mew manga ends with a "wedding" ceremony between Ichigo and Aoyama-kun.
An episode of Kaitou Saint Tail had the title character go after a stolen wedding dress design at a fashion show by sneaking in as one of the models. This also gave her best friend Seira, acting as her decoy, an identity crisis of sorts — she's a nun.
Lampshaded in Lucky Star, where Kagami ponders on the religious ramifications on her sister wanting to have a Western-style wedding.
In YuYu Hakusho, it is revealed in a flashback that Kurama's human mother Shiori had a Western-Style wedding.
The very last page of the last chapter of Fruits Basket has in the background what is most likely Kyo and Tohru's (or their son's) Shinto wedding photo.
The June episode of Bottle Fairy had the four fairies imagining their prospective weddings to Sensei-san.
In Code Geass, Ohgi and Viletta get both Shinto and Western wedding ceremonies. Kallen attended both ceremonies and has photos of both in her bedroom. Lampshaded in an official picture where the groom wears a wedding kimono and the bride wears a Western bridal gown, which makes sense, seeing as the where the bride and groom are from.
At one point in the Ah! My Goddess manga, Keiichi had a dream about a Western-style wedding with Belldandy. Tamiya was the priest, complete with skullcap and pectoral cross, and Otaki was the altarboy.
In the first instance, all the dead Suzaku Seishi return for the final showdown with Nakago, and the only one missing is Tamahome. It suddenly cuts to a sequence where everyone is preparing for Miaka and Tamahome's wedding (which was about to take place in a Catholic church), showing up in tuxedos. After a conversation with his father, Tamahome (who was himself wearing a white tuxedo in this sequence) learns that it was all a dream and resolves to go back to fight Nakago.
It happens again in Eikou Den, where Miaka and Taka are finally married in a traditional Western-style wedding in a Catholic church. Contrary to the dream sequence, Taka was wearing a black tuxedo, underlining the difference between him and Tamahome.
In Hell Teacher Nube, Miss Ritsuko is possessed by an obake that makes happy brides out of its "victims" (it's generally a benevolent, well-meaning spirit who helps couples tie the knot.) Nube is ecstatic, of course, and they very nearly conduct a traditional Shinto wedding... until Nube realizes that Ritsuko only agreed because she was possessed, and slaps the obake right outta her with his sutra.
The rereleases of Chrono Trigger featured a Western-style wedding in one of the cinematic anime endings. Justified since the characters involved are from a Western-style medieval setting.
Same applies for the weddings of Gale and Sakura (Haru's parents) and Haru and Elie in Rave Master, whose world is based more on the western world.
Several Super Sentai series have episodes that revolve around the Monster of the Week targeting brides and/or the heroes staging a fake wedding to entrap the bad guy, almost always in western-style ceremonies. Even the rare hero weddings that are entirely legit, such as in Mahou Sentai Magiranger, tend to be western-style.
Samurai Sentai Shinkenger had a mix of both types in its bridal episode, with Mako and Ryuunosuke disguised in the Shinto wedding kimono.
An early Sailor Moon Episode involved a monster targeting a Bridal Fashion show, and Usagi constantly fantasies about a Western Wedding (with a bunny stain-glass window). In the manga and live-action show, Usagi and Mamoru end up having a Western wedding, complete with a Catholic church.
Bakuman。 has a seemingly Western-style wedding in which Akito Takagi wears a tuxedo while his wife Kaya (who had legally married him a little while before, and scheduled the ceremony for when they had time), wears a wedding dress. During the wedding, Akito's partner Moritaka's mind is mainly on the question of whether they should stop working on Tanto.
And the cover art for the final chapter indicates that Mashiro and Miho opted for a Western-style one as well.
Recent Kamen Rider has had a few more weddings than usual: Nago and Megumi in Kiva, Akiko and Ryu in Double, and in Movie Wars Core Eiji and Hina from OOO pose as a bride and groom to play roles in a movie. All these weddings were western-style.
Photographs in Tiger & Bunny show that Kotetsu and Tomoe had a Christian-style wedding, which wouldn't be remarkable in the anime's New York-influenced setting were it not that the Kaburagi family is depicted as observing many Japanese cultural traditions, including Tomoe's Buddhist burial.
In Slayers, Martina and Zangulus have a western-style wedding at the end of Slayers Next. However, Slayers itself takes place in a Western European setting, so a western-style wedding is only logical.
While Ureshiko in Oku-sama wa Mahou Shoujo wanted a western-style wedding, she opted for a Japanese one. Reason being, the western ceremony requires a kiss, and if she kisses someone, she'll lose her powers.
In Rummer Godden's In This House of Brede a group of Japanese girls come to an English Benedictine abbey to go through the novitiate and then return to found a branch house in Japan. One of the girls wants to have two bridal gowns—one Japanese, one Western—for her 'clothing' (the ceremony where she is symbolically 'married' to Christ); the Abbess & the Novice Mistress talk her out of it.
In My Forged Wedding, Takao's route ends with a western-style wedding, which neither of the participants consider in any way binding since they're not planning on filing a marriage license and are just doing it to make Takao's grandmother happy.
The "Wedding Kiss with Yamato" side story also involves a western-style wedding, this time a mock ceremony being put on to promote a wedding planner's services.