in that case, I think they have to put a little wrapper in front of their ReST API, because DW WorkflowManger cannot handle ReST URLs directly. It uses an ASMX service. You can pass it some of your index data or workflow variables and it may give you a return string back, too.
To demonstrate the ASMX with Csharp:
public string HelloDocId(int DwDocId)
// do your ReST API calls here ...
return "Hello Doc ID: " + DwDocId.ToString();
You can compile more than one web method into a single ASMX webservice DLL for the IIS. This example will get the doc id number as an input parameter from workflow manager and returns a result string back to workflow manager. So an ASMX web service is quite simple thing. Instead of the "hello doc id" line your customer's invoicer has to put all his necessary ReST calls in this method, that's all.
Because of they already could provide you with a public reachable URL, they just have to do the same with the ASMX wrapper service.
But be aware of security: If the service can be reached without any restriction from over the internet, everybody who openes the URL in the browser can pass some random parameter values into your service (i. e. docid 4711 and clicks OK). Then without any other security mechanism like a simple passphrase as a second parameter for example, your invoice system may start booking/printing/exporting/... doc id 4711 automatically and will do everything just like your workflow manager has triggered the service! Consider your invoicer to restrict that service to your workflow manager server only, too.