Put the technical issue aside for a minute: There's an ethical consideration here. If your website collects credentials for another website, you can log these credentials and use them for evil. There's even a term for this; it's called phishing
. Whether or not your client has any intention of abusing the credentials they may or may not collect, it's bad to put yourself into a position where abuse could happen. So I would side with the developer purely on ethical grounds, and decline to build such a feature.
Technically, you might be able to send a POST request directly to Hotmail containing credentials, but then what? You redirect to hotmail? Why not just have a link to hotmail and let the user enter credentials once they land there? A website as huge as Hotmail probably blocks such attempts anyway.