I had problems recently with my iPhone 6 vibrating randomly. I didn’t get any alerts, banners, or app icon badges that I could see, so it was driving me a little crazy.
It turns out there are two kinds of mail notifications. The first kind is under Settings > Notifications > Mail. The second, which causes these phantom vibrations, is under Settings > Sounds > New Mail. Head there and change “Vibration” to “None.”