Sorry to disappoint you but those providers may have lied to you. MMS, as a protocol, uses mobile data. It's just part of it. Just like you can't get an SMS without a mobile signal because of the protocol, the same thing applies to MMS. You can check it out in your own phone. Connect to WiFi, make sure the 4G/3G icon goes away. Now send yourself an MMS from another phone. Voilá! 4G/3G icon comes alive, downloads the message, and then goes back out.
HOWEVER, some providers do circumvent this with re-routings. Verizon, for instance, lets you use their own app that re-routes SMS/MMS to an IM protocol which can be sent/received via any kind of data, including WiFi. But again, that needs you to use a special app. That's also similar to how WiFi Calling does it, because it routes all your call/data/messaging through WiFi, but for this both your phone and your provider need to support it.
And yes, the Advanced Messaging protocol that T-Mobile announced it's supporting this year is different and will have different protocol rules. But that's something for the future, not right now.