Can I set the iPhone to vibrate except for one contact?

Solution 1:

Buy a 'silence' ringtone from the iTunes Store. Make this your default tone, except for that one person.

Solution 2:

You cannot do this, as you cannot define exceptions to the Mute toggle. Here are the possibilities I see, none of which solve your problem exactly:

1. Disable all sounds for all apps, including ringtones (you can do this by making a blank ringtone file yourself, or by getting one from the iTunes Store). Then go into that one contact and assign a ringtone to them. However, disabling app sounds also disables their vibrations.
2. Using Do Not Disturb allows you to specify a group of contacts that will bypass the setting, causing your iPhone to ring. You can find this setting under Settings → General → Do Not Disturb → Allow Calls From. Again, this will also disable vibrations, as you appear to know already.
3. In the overkill category, you could get an Apple Watch. Notifications will transfer to your Watch and not ring or vibrate your iPhone. You can then disable sounds without disabling taps on the Watch. When you get a phone call, your iPhone will still ring (this is the one exception to notifications on Apple Watch). From there, disable all ringtones except for that one contact as I described in 1.