Brandon,
I don't use Story Maps, but as far as I understand, when you download the example and host it on your server, you should be able to edit main-app.js
Here's a reference example to the widget in 3.20, the version the story map example is using:
Basemap gallery | ArcGIS API for JavaScript 3.20
I would add the reference to
esri/dijit/BasemapGallery
Then instantiate it:
var basemapGallery = new BasemapGallery({
showArcGISBasemaps: true,
map: map
}, "basemapGallery");
basemapGallery.startup();
You'll need to add a front-end reference to the basemapGallery div, probably to index.html