Hi Cole,
Q) Using Portal for ArcGIS 10.3.1 and WAB Dev 1.2, which has the analysis widget, can you use this widget when deployed to the web server?
A) No. The Analysis widget in Web AppBuilder Developer edition v1.2 is NOT supported with Portal for ArcGIS 10.3.1. The widget only works with an ArcGIS Online organization.
As Jian said, support for the analysis tools in Portal for ArcGIS will be coming in the 10.4 release - which is still in development.
Hope this helps,