Your architecture looks well.... at least strange. But let's skip that part.
A service is most likely running under one of the system users. That one has no profile, but even if you use a dedicated user, the service is running without profile. So starting processes like this won't work. Besides, if you have Vista or above, it runs in Session 0, in which starting interactive applications might result in unexpected outcome.
But if you want to issue a http request from a service, you don't need to (actually you never should) do this using a browser. For that there are tools in .net:
WebClient[
^] and
HttpWebrequest[
^]. Use those instead they will work. You will need to refine your code, but what will work in Windows Forms or as console, will work also in a service. But be aware of the running user if you need authentication.