I haven't heard about shinto style wedding but I guess it's shinzen-shiki.
Is it the wedding with kimono?
Here is some places you can have shinzen-shiki.
Maybe you need to ask the staff if you can do the wedding for just you two.
I recommend you to do the wedding at shrine because it's like more Japanese traditional.
Asakusa shrine
03-3844-1575
Ikenobata bunka center
03-3822-0201
Wedding reception house Raphael
03-5790-7551
Aoume sweet plum
0428-23-2155
Ookuni tama shrine
042-362-0092
Oomiya hachiman-gu(shrine)
03-3312-7515
Kanda myojin (shrine)
03-3254-0753
Ginza blossom
03-3543-5677
Tokyo daijin-gu Matsuya salon
03-3234-6611
Nedsu shrine
03-3822-8753
Hie Akasaka(shrine)
03-3502-2205
Meiji kinen-kan
03-3403-1177
Meiji shrine
03-3379-5511
In a case the staff doesn't speak English, here is some examples of Japanese sentence that you can use.
Can I do shinzen-shiki?
Shinzen-shiki ga deki masu ka?
Can I do ceremony only for us?
Watashi tachi hutari no shiki wa deki masu ka?
If they say ''deki masu'', it means ''yes''.
How much does it cost?
Ryoukin wa ikura ni nari masu ka?
Please tell me the total price.
Goukei kingaku o oshiete kudasai.
When can we do the ceremony?
Shiki wa itsu deki masu ka?
Please speak slowly.
Yukkuri hanashi te kudasai.
I hope you have a good ceremony! Good luck!
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