Hi Aaron,
ArcGIS Online uses the World Route service, which cannot be changed by users. Currently, with ArcGIS Online you cannot specify your own Routing Service (you can specify your own print and geocode services, though). If you are using Portal for ArcGIS (free if you have ArcGIS Server Advanced or Standard), you can configure the routing service with your own published route service.
Another option would be to create a custom application using the ArcGIS API for JavaScript, and use your own published routing service. Here are some examples:
Find routes with barriers and multiple stops | ArcGIS API for JavaScript
Find a route | ArcGIS API for JavaScript