Cannot save image files into their originating folders using the 'batch' command.

"

I leave the destination as 'none' which may be where I am making the mistake.

When i do all of this, the software just saves all of the WEBP files into one directory. "

Yes, that is "mistake"you have recorded saving path (destination) in your Save step without overwriting it. See explanation provided by @Stephen Marsh 

Batch will open the files, so all you need is the Save As step with the Google Photoshop WebP plugin settings as per the attached screenshot annotated with Snagit:

Here you can see the result, the original files were not altered and the WebP versions were saved in each sub-folder under the main top-level folder next to the original files:

More on Automate > Batch here:

https://helpx.adobe.com/photoshop/using/processing-batch-files.html

The Image Processor and Image Processor Pro scripts can also save into the originating folder.

I was pleasantly surprised with the support for both actions and scripting with this plugin:

Choose in Batch destination Folder and override Action location.

Make sure you are not including a save path in the action - it should just be Save - OK.

You should also make sure you're not doing anything to the file that is not allowed in the jpeg specification. That is, pretty much anything. The file has to be flat, 8 bit, no transparency and no alpha. The safest way is to include these steps in the action to ensure jpeg compliance.

As long as you have a Save in the action, batch destination should be "none".

WEBP is a dark horse here. The format is not natively supported in Photoshop, you need a plugin. So maybe that causes hiccups.