Hotkeys are registered by the running ZunShot instance. If another program owns the key, or ZunShot is not running, nothing happens.

Look for the tray icon. Closing the main window keeps ZunShot in the tray, but Exit stops the hotkeys. Turn on Enable screen capture when Windows starts in Settings > General to have it ready after sign-in.
Only one program can register a global hotkey. Change the key in Settings > Hotkeys, or close the other program. A second ZunShot instance also counts.
Many laptops need Fn+Scroll Lock or have no key at all. Use Alt+PrtScn, which is also assigned to Instant Capture by default, or assign any other key.
Windows can reserve PrtScn for its own Snipping Tool. When you assign PrtScn in ZunShot's settings, ZunShot asks for permission to turn that Windows option off and turns it back on when you stop using PrtScn. In the Microsoft Store version turn it off yourself: Windows Settings > Accessibility > Keyboard > Use the Print screen key to open screen capture.
Games running as administrator do not receive hotkeys from a normal process. Run ZunShot as administrator for that session, or use windowed mode.
Default keys are listed in the guide under Settings - Shortcuts.