Simplest solution is to use ArcGIS Online, they allow you to upload and plot your shapefiles.
The other route you could go is use an intermediate service (with ogr2ogr) to convert from your shapefile to GeoJSON. Something like:
ogr2ogr -f GeoJSON data.json data.shp
Once you have GeoJSON, you can plot that on the map... though Esri needs ArcGIS JSON instead of GeoJSON. Esri has already taken care of this for you with the geojson-layer component in their github repo.