I noticed that the Browser category in the ASuite menu is missing now. It was there yesterday.

How would I go about restoring it?

Are you using both old (Liberkey 4.9) and new menu (Liberkey 5.0 beta) ?
If yes, you maybe moved it into another category. Try to find it using the search engine.

If not, you can use ASuite.xml backups into Liberkey\Apps\Asuite folder (named 00x.bak)

No, I'm not using the 5.0 yet.

I went into the Liberkey menu but I don't see anything in the liberkey tools submenu for backups. Will I need to reinstall the menu and if so, will that kill off all the entries I already have?

Look for Asuite.xml backups (.bak) into Likerley\Apps\Asuite folder