Not directly, no. The app has exactly the same rights as its user has. That's part of the OS's security model. If I had to deal with something like this, I'd do it this way:
Create a second program that runs as a Service, and set it to run under a user account that has access to the shared folder. It should implement some sort of validation logic, and listen for incoming messages. (What exact method it uses for this is up to you, but you're essentially creating a server.)
Your desktop app runs under the limited user accounts. To request a file, it sends a message to the server, in which it identifies the user and the request it's making.
The server checks the request, and if it's valid, retrieves the file and passes it back to the user app. If not, it should return some sort of error message.