I've attached an image of the code where the issue occurs and the errors below opened up. It is the highlighted line that fails.

So in the line that is failing you are trying to create a reference to the LfVolume but have the LfConnection and LfDatabase been properly instantiated?  Here is a code snippet from the LFSO documentation;

' Creates a new application object.
Dim app As LFApplication = New LFApplication
' Finds the appropriate server.
Dim serv As LFServer = app.GetServerByName("MyLFServer")
' Gets the repository from the server.
Dim db As LFDatabase = _
  serv.GetDatabaseByName("MyRepository")
' Creates a new LFConnection object.
Dim conn As New LFConnection
' Sets the user name and password.
conn.UserName = "myusername"
conn.Password = "mypassword"
' Connects to repository.
conn.Create(db)

  (If you leave the UserName and Password properties empty the connection will try to login using WindowsAuth)

Once the LfDatabase is properly instantiated and connected then you can get a reference to the LfVolume with this code;

LfVolume vol =  db.GetVolumeByName("myVolumeName");

Yes. I have attempted both with and without the username and password. I checked the SQL users and both seem to have the rights to the database. I think it can see the server because when I didn't set it up properly before, I didn't see this error, but I am not sure at all.

A couple of thoughts;

• How about checking the InnerException property of the Exception error that is being thrown to try and get additional information? 
• Can you code up a very simple test app using the code snippet from the LFSO documentation to test your connection?

BTW - The credentials that you need to supply to the LfConnection object are  Laserfiche user credentials not SQL credentials.  For testing purposes you could use the same credentials that you use to open the Administration Console

I'll try out setting up a test app. The inner exception appears to be the exact same thing. So the credentials should be the same credentials that I use to get my license key?