Although I suppose there might be a legal issue I don’t see a moral one at all - what if you built a fully working human android and sat it at the computer, watching the screen for you? or you paid a real human to do it for you? I don’t see how that could be immoral. It’s only money anyway.
In general, when companies expose functionality on the web, I don’t think they get the right to say what sort of browser you use to access it with and I would see this as just ‘extending the functionality’ of your browser.