iPhone vibrates several times a day with no visible notification

Check your mail notifications specifically: it is possible to have vibration without sound or other notifications on a per account basis.

Settings > Mail > Notifications and then traverse each account. Make sure both sound and vibration are both set to none, if that’s what you want.

For me at least, this ended up being the "Airmail" app, which I use instead of the default Mail app. I had notifications set to Badge, but due to how Airmail retrieves messages, the received date of emails would almost always be many minutes behind when the badge occurred. Oftentimes substantially, so it never seemed to align.

The app gives no insight into how it retrieves messages. I'm assuming they are merely fetched at an interval and not pushed, which if they were would likely avoid this confusion.

I eventually confirmed this by disabling notifications altogether for several apps, and narrowed it down to Airmail in the end.