As usual, I should have given myself 10 more minutes of googling before I posted my question.
Well, for anyone else wondering how to do this, here is what I discovered that works:
'Read the file into a byte array
Dim saleBytes(uploadedFiles(0).ContentLength) As Byte
uploadedFiles(0).InputStream.Read(saleBytes, 0, saleBytes.Length)
Dim s As String = System.Text.Encoding.ASCII.GetString(saleBytes)
'Now "s" contains the exact contents of the CSV file, or whatever may have been uploaded
Edit: Sorry, when I click the "Format Code Block" button the WebSense filter at work blocks the resulting page. Hopefully the code is readable without it.