I cannot find my desktop account online. Within the application, it shows I am connected. Still, when I try to login via a browser, I show no purchase.

You are asking this in the actual "desktop" version area. Desktop is QB Pro, Premier, Enterprise that you download to your own computer and run without any internet connection except for updates. It is not clear what you think you purchased.

Log in to your actual INtuit account at camps.intuit.com to review your purchases of any Intuit product. This includes TurboTax, QBO, QBDT, Mint, etc. You should only have one intuit account fro everything