I Was Today Years Old When I Learned You Can Scan Documents on an iPhone Using the Notes App

Once you've clicked the Notes app icon and you have a new notes file open, locate and tap the "+" icon that can be found on the bottom middle of your screen. If your keyboard is on, it will appear in a bar right above the keyboard.

Note: If you don't see it, go to Settings->iCloud->Notes and turn that on. Then, once you go back to Notes, the "+" should appear.