Is it possible to restrict the mod to a certain permission group on a server? Or can I only restrict it to admins/operators?

Fabric doesn't have a concept of permission groups, but you can disable the mod for anyone without OP by disabling the menu for players. I could add an option to only enable the menu for operators, if that is what you are looking for?

I meant more from an operator's standpoint on a server: would I be able to restrict use of the mod to anyone outside of a permission group, making use of a permissions plugin, or is that not possible with the mod? I am not sure how well fabric works with plugins or stuff like that.

If a permission plugin existed, it would be possible to provide compatibility with additional work, but the current phase of the project doesn't support anything like that (because Fabric does not have ""permissions"" built-in; it would have to be a 3rd-party library with external capabilities, and I can't name any libraries/projects that fit the bill for that).

If anything, it would be a separate extension mod built specifically for Identity that hooks into permission mods. I would be happy to look into it if you have a particular permissions library/system you are already using.