Yes. Moodistory supports your device’s Face ID / Touch ID / Passcode to restrict access.
Just go into Moodistory’s settings and enable “Touch ID / Code” or “Face ID / Code” or “Lock” (depending on which device you use).
Once enabled, Moodistory will ask for your Face ID / Touch ID / Passcode / Unlock pattern when opening the app and before anything else is shown on the screen.