Hello, thank you for reaching out to Microsoft Q&A.
May I please know which version of the Snipping Tool app is installed on your device?
(Go to Settings on the Snipping Tool app > See which version is currently installed).
If you have version 11.2601.20 (latest version) installed, you should see the option to record video, as shown in the image below.
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If this option is missing, just uninstall and reinstall Snipping Tool on your device.
Search for Snipping Tool in the search bar on the taskbar, right-click the results, and click Uninstall.
Once uninstalled, search for the Snipping Tool in the Microsoft Store to reinstall it.
Let me know if you still need more help.