Creator of theq extension here, that is an optional permissions (may ask). It never asks to access to all sites, it only optionally let’s you add support for additional sites such as different Invidious instances. These are arbitrary domains, so it needs permission to ask for any arbitrary website.

Support for invidious is not actually done yet, so right now, it never asks for permission for any new sites anyway

The client side data is used to store your submissions to always show your preferred title, even if there is another submission on the server. You can disable this option in the settings