Thanks for the suggestions Joey. I added a write to file task with a blank line in between the Read and Output task, and it gives me this:
StudentLastName, StudentNumber, HomeRoom
Smith, 12345, Blue
Johnson, 54321, Blue
Jones, 78941, Green
Harmon, 54891, Green
Which made me laugh because that's the behavior I feel the append to file task should be working (i.e.starting on a new line).
I did not want to modify the SQL script, but I can try that as a last resort.