How do you upload a web map in our app in MIT App Inventor?

Consider making a FeatureCollection of your ArcGIS map. Load load the FeatureCollection to your Project using the MIT Map component.

A tutorial showing one way to make a map overlay is in two parts

This is a 'work around' that might work for your Project that will work with the LocationSenser or without.

Evidently it might be possible to export your ArcGIS map directly as a FeatureCollection