Is there a way to delete all items in my Stack Exchange inbox? There doesn't to be a 'clear inbox' button.
You can't. There's no way to delete old items from your inbox.
However, as soon as you click the top bar icon to look at your inbox, all past items in it will be marked as read, and the red inbox alert icon will disappear until the next time you get a new message in your inbox. (If you have multiple tabs open, the alert might persist in other tabs until you either click it or refresh the tab.) You can still open the inbox menu again to browse and click on the old inbox items, but they won't be highlighted as new any more.