I do almost all of my work from an iPad as it functions, in most cases, almost identically to a computer in terms of end result, but I've noticed a lot of talking about the desktop app. Is it required to work through the desktop app, or can I do everything necessary through the website itself?
The desktop app is NOT required.
Many Upwork freelancers do NOT use the desktop app.
Simply using the Upwork website, you can do everything that the desktop app does, with only one exception (logging time on hourly contracts automatically).
But if you are doing fixed-price work or manually logged hourly work, the desktop app really is unnecessary and irrelevant.