The WSDL (Web Service Definition Language) is an XML file that documents request and response types as well as the remote method signatures. Read the WSDL and write the method stubs. If you are looking for a code generator to produce method stubs from a WSDL, then do a Google search and find a utility that best fits your needs. Adding a service reference or using the wsdl.exe or svcutil.exe will generate the request/response types. Just copy the types to the mock service project.
What is the current technology?
WSDLs document SOAP (XML) services.
What would I have to request from the customer? Which file?
Again the WSDL file. Service owners generally expose a testing site. Contact the the service owner for support as they know their systems and can guide you far better than a ASP.NET support forum.
How can I easily include this in my client C# desktop application.
It is a matter of learn the fundamentals. If the service owner has a test site then create a service reference as explained in your other thread using Visual Studio. If the service owner only provides an out-of-band WSDL which is atypical then use the wsdl.exe or svcutil.exe to generate the C# source. Both utilities let you select an output location. Read the documentation!
Creating the Web Service Proxy
ServiceModel Metadata Utility Tool (Svcutil.exe)
Adding code to an ASP.NET Visual Studio project is very easy. Just copy the files using Windows Explorer to a folder in your project or if you used the code generators make sure the output is a folder in your project.. If you cannot see the files in solution explorer, then right click and highlight Add -> Existing item... Find the file(s) and click ok.
For ASP.NET Core projects just add the files to a folder in the project.