Many thanks to Jonathan Chang for getting me most of the way to this goal! I am sharing some details of the process of identifying the URL here. Check out the linked tutorial; I am illustrating some of the steps to get to where you can find the API.

1. Right-click on the Story Map web page and select “Inspect”
2. Select the “Network” tab in the console
3. Filter by “Fetch/XHR”
4. Refresh the page
5. Scroll down to the layer you want, and watch for “FeatureServer” in the “Name” field
6. Right click this entry, “copy URL” and paste into the browser, and delete the string including and after the ?
7. This should get you to the API where you can see the layer list; is your layer of interest there? If so, it is followed by a number in parentheses. Click on it, and this is the URL you can use with the esridump tool