Most likely: Yes. They can ask. Well, in fact, we know they can ask. Your question seems to be whether they can ask on a site that may not be secure. Probably yes.
You can decline to provide that information online, as you did. And they can decline to accept you.
And more specifically: How did you plan on sending that information? By mail? Fax? Are those any more secure? Not necessarily. And that only protects the transmission. Once your information arrives--whether by a web form, mail, or fax--you have no control over how the recipient will handle or protect it.
I'm guessing that they have a way of doing business, of processing applications. You're asking for an exception, a different way of handling that. At the very least, you've disrupted their process. At worst, you've pegged yourself (in their eyes) as a trouble-maker--someone who's starting raising issues even before the application's been approved.
Understand: I appreciate your concern. It's not out of line. But they have their way of processing applications, and you've decided to seek another way. There's a risk to that.
Hope that helps.