With the current implementation of MapBox, there's no way to pass the access token, which, unless I'm mistaken, is necessary when not using the example map ID (as used in the Statamic docs).
If you register an account and create a map, then paste the map ID into the
mapping_service_api_key as instructed in the docs, then set up the template tags correctly, you just get a blank rectangle with the markers in the correct locations.
Following this example, however, renders the map. If you're logged in, it'll provide you with your access token right in the sample code.