How can I find my AirPod case without the AirPods being inside

You can't, not even if the Airpods were inside.

To learn 'How to find your lost AirPods, AirPods Pro, or AirPods Max,' click here --> https://support.apple.com/HT207581

Using the features offered by Find My is the only way Apple offers for tracking or locating a lost or missing device. There is no other Apple service that can find, track, or otherwise flag your device for you. If they do not work then you will have to resort to conventional methods for locating lost items. For Find My, ALL the requirements in this list must be fulfilled for Find My to even have a chance to work:

- Find My [device] must have been set up before they went missing.

- They must be in range (<~150') or all you will see is "last location".

- They must be charged (see the link for details).

- They are not inside their Charging Case (Airpods Max Smart Case is not the same thing.).

- They have not been reset by somebody who found them.

- To find a single Airpod, put the one you have in the charging case first.

- You cannot track a lost case with Find My.

Airpods won't show in Find My if they were not set up in Find My first.

The support article also tells you how you can order a replacement.