I had an issue where *ONLY* MMS messaging was not working - SMS worked just fine.
As only one protocol was not working, this led me to believe it was a Verizon Network issue and not a device issue - which made sense as literally every other service was working just fine.
Verizon L1/L2 tech support never thought to ask if I recently switched SIM cards - they also never asked about APN settings (past or present - this is probably a pretty small edge-case)
My specific issue is because my phone is unlocked and I was international using a different foreign prepaid SIM (which had specific APN configuration for the foreign network). For whatever reason, even after a full factory reset and Network Settings reset, the APN data from the previous SIM was still stored (somewhere, can't tell you where) in the phone - to resolve:
1) Popped the old international SIM (I honestly don't know how you'd solve this if you didn't keep the SIM card)
2) Went to Settings
3) Go to Cellular
4) Find "Cellular Data Network" (this option is not available with Verizon SIMs - it really should be and not hidden)
5) Remove all values AND click Reset Settings (or some combination) - I removed the values and then clicked Reset Settings and then left the page
6) Pop back in Verizon SIM card and let the phone re-active
7) Reboot phone for good measure
No more MMS issues - all images go through and group chats with MMS works fine
Verizon should really be aware of this issue as I can see anyone traveling internationally could face this - the L1/L2 teams are not stood up to check this use-case (which, I admit, is slightly edge-case). However, I'm not sure why the Verizon SIM doesn't over-ride or reset these APN values when a new SIM is detected. I also don't know why Apple iOS doesn't reset these values when doing a full factory reset or a Network Settings reset.