Hi Tim,
> I do not have ArcGIS Pro. Is this required? If not, what do I need? I need to publish not on AGOL, but WAB DE using the LocalLayer Widget. This is not going to be shared with the Public.
Yes, ArcGIS Pro is required. It is not included with ArcGIS 10.2.2, but it is part of ArcGIS Desktop 10.3 and later releases. You need ArcGIS Pro to publish/share a scene layer. A scene layer only lives in a "portal" and not in a GIS Server. In the GIS Server, it is called a scene service. It is NOT possible to do this with ArcMap. In order to "store" a scene layer, you will also need to enable either an ArcGIS Online organization, or a Portal for ArcGIS instance with a hosting server configuration and the ArcGIS Data Store.
FYI, help topic: Scene services—Documentation | ArcGIS Enterprise
Hope this helps,