To summarize the places to look:
Mute: either in the Control Center or the side switch, depending on your what you have set in Settings > General > Use Side Switch To:
Sounds: In Settings > Sounds you choose what sound the arriving and sent email makes. They can be set to None.
{thanks to Johanna for those}
Notifications: Under Settings > Notifications > Mail you can turn off notifications altogether, or you can turn off notification sounds for individual accounts; by selecting those accounts.
{thanks to abababababab}
Do Not Disturb: If you have Do Not Disturb turned on under Settings > Do Not Disturb then most alert sounds will be turned off. There should also be a moon icon in the status bar.
If none of these work it might be time to try Settings > General > Reset > Reset All Settings. This wont' erase any apps, data, or accounts, but it will default many other settings, like wallpaper and sounds. You'll have to tweak some of your preferences to get things back to where you like them.