I went through forum after forum on this issue yesterday looking for the answer. MMS would not send unless I was on wireless. WiFi would not send them at all. This was tried with the Samsung Messages and Google Messages apps. This is a very common issue and has been documented on google very well.
There are many potential solutions to this issue but I have on simple thing that you should try before you waste time with resetting your network, clearing your cache partition in recovery mode etc..
Use the Verizon Messages App! That is it.
Samsung messages and even google messages no longer allow you to send MMS through WiFi. Even when you tell the system to stay active on wireless data from the Developer mode. The only app that will work for this on Verizon and fixes it, is to use the Verizon Messages app. This app will work for SMS/MMS on WiFi or wireless. Instant fix!
Oh dear, I am sure you would like to get your pictures sent out to your friends and family, Robertsd0275. I would love to help in seeing what is going on. I do want to clarify a few things. Is anyone else having issues with picture messaging? How is your text messaging? Are you able to use your data? -AmberF_VZW