Your contractor is wanting a REST link that you can provide through ArcGIS Online. Almost everything you publish on ArcGIS Online has some kind REST service, but there are a couple caveats.
If you have publish a layer to ArcGIS Online it will say Feature Layer (hosted)

If you got the items details page at the bottom right column will be a REST end point service link.

For Feature Layers it is a Feature Service; if you publish an entire map, it will be a Map Service link. Do note that ArcGIS Online Map Service links do not support Query operations, so that might change your use cases.

Just change your sharing options to ensure that the contractor can see the layer, if it is a layer with editing enabled, create a VIEW of the layer and share that instead. If it is something that they need to edit, do realize that the edits will only exist on ArcGIS Online and not your local file geodatabase (FGDB).
I think I might be going crazy unless ESRI changed something, which is VERY likely. I am 99% sure, you used to be able to create service links for host Maps. But it looks like you need to turn the Map into a hosted Tile Service now for a Map Service link.