No. Because your background process on your computer can't possibly be running any of the forms running on someone else's form they created and are running on their computer using Visual Studio C#.
Now if your question was can you WinForms application just use a single background worker process, yes, you could. But then you'll now have to figure out how to queue work to your background worker process so that it can handle multiple tasks should you have multiple background tasks to do. You'll end up having to write a mini-operating system. Typically, people will create a background worker that is just focused on a specific task so as not to have to solve that multiple tasks but only single process problem.