The confusion seems to be whether the OP can send MMS while connected to wifi. The answer is yes. I have absolutely no problem sending MMS messages and I am always connected to wifi at home. The iPhone should automatically send it via your carrier, even when connected to wifi.
Exactly.
But if you dont have carrier data on your iphone or the correct apn settings installed for MMS it will not send the MMS over wifi.
It strictly uses carrier data to send mms and it works fine to send while you're connected to wifi. It just doesnt use the wifi connection, it sends it with carrier data instead.