Did this start happening recently after an update to the phone or has this been happening for some time?
There are a few more troubleshooting steps that can be done, hopefully we can narrow this down before a factory reset is required.
Boot to safe mode:
Turn your phone off.
Hold down the Power button until you see the Galaxy S7 logo appear.
Immediately press and hold the Volume Down button.
Once you see "Safe Mode" in the corner of the screen, release the Volume Down button.
Test to see if you're able to send yourself a MMS message.
If the above doesn't work, the next step is to delete the phone's cache. This only holds temporary information from apps and the system, and will not erase any of your data:
Navigate to Settings > Device Maintenance > Storage
Tap on the menu button in the top right and tap on Storage settings
Tap on Internal Storage.
Tap on Cached data.
When prompted with "This will clear cached data for all apps", tap on Clear.
Next, turn your phone off and wait for it to shut down.
Hold down the Volume Up, Home and Power buttons at the same time. You will see the Galaxy S7 logo appear at which point you can release the Power button, but keep holding down Volume Up and Home.
You'll see a screen displayed that says "Installing system update", followed by an Android with an exclamation point which will be displayed for about ~30-45 seconds, after which you will see several recovery options.
Use the volume down button to highlight "Wipe cache partition" and then press the Power button to select it. You should be asked to confirm, use the volume key to highlight Yes and then Power again to begin the cache wipe. Once it has completed, use the Volume Up key to select "Reboot system now" and then press Power to restart your phone.
Once your phone has restarted, test MMS again.
If the above doesn't work, you will need to factory reset your phone. Before doing this, make sure that you have saved any important stuff like your photos, contacts, etc. Once you've got everything backed up head over to Settings > General Management > Reset and select Factory data reset, and follow the onscreen prompts. Your phone will restart and begin the factory reset process, and once it's completed you'll need to go through the setup process (sign in with your Google account, Samsung account if you have one, etc).
