Ok now I got you, and that makes sense. It is actually a common scenario and I too have come up against that. There are several ways to go about asking System 'A' (the vendor system) to provide you with data, so that you can use it in your system 'B' (correct me if I am wrong on that statement).
At this point is nice that your vendor will actually work with you on how you can consume thier systems data; normally it is more of a dictatorship and it is told to consumers how to get the data and that is that.
The 1st (and maybe not reccomended, but used often) way of getting that workflow data would be for the vendor to export thier data into either a flat file or XML file (preferrable) that you could import when needed to your system. You could automate bringing in the data into your database or directly to ASP.NET via FTP, a SQL Server Job, or a Windows Service that monitors a directory with a FileSystemWatcher and when a new file is added, the service runs and imports the data. The other difficult issue with this methoid is the communication is mostly 1 way.
The 2nd (and more preferred) would be to build a new intermediate service (WCF or .asmx) that allows your app to call, and the vendors app to respond with the data real time. A very basic example of this would be how some weather services work. Think if I built a custom .NET app that wants to display weather info from a vendors app (i.e. weather.com). I would hit thier service requesting the data, and weather.com's data (or vendor data for your example) would be passed back for my app to consume. In your case you could talk back and forth via the service to complete the process since some of system A's process can not be completed until System B has done some work as you mentioned.
I would not be intimidated by creating a service for the apps to communicate dat back and forth. It will just be up to you and the vendor to agree on the way the service is built and consumed, so you can pass data to each other.
Other than that I don;t think there is a simple 'link 2 web apps and go' solution and using the querystring might work if you have a secure handshake between the (2) sites, but if the data is too large that option is not as workable.
I hope this offers some help to you!