It seems that it is not possible to load data in a Graph using the "url" parameter on a private wiki. This happens regardless of whether the source is on the same wiki or external.

This is inconvenient as it forces us to embed all the data in the graph itself. As for maps, it would be silly to duplicate all the basic geodata in every map, so we can't really create maps right now.

An example is available at https://office.wikimedia.org/wiki/Template:Graph:Map , which contains the same content as its (working) counterpart on the English Wikipedia. The console indicates the error:

load.php?debug=false&lang=de&modules=ext.graph.vega2&skin=vector&version=1qq0zep:448 [Vega Err] PARSE DATA FAILED: countries Error: API error: {"code":"readapidenied","info":"You need read permission to use this module","docref":"See https://office.wikimedia.org/w/api.php for API usage"}

The same error is visible in the GraphSandbox when trying to load other types of data using "url".

This seems specific to private wikis, so sadly I can only provide an example with restricted access.