If I'm remembering this correctly, I'm pretty sure that it was Ein doing the heavy lifting:
Anyone who hooked into this website got brainwashed (Ein had saved Jet from this by biting him), so I guess they figured if a Dog's brainwaves were used, the website won't be able to brainwash them. So it's Ein that's doing the controlling (you can see him twitching and moving) and it looks like everyone else was just monitoring his progress:
When the site comes up, Jet says "Ed, that's absolute perfection!", thinking that this was Ed's doing, but then it pans to Ed in front of her computer confused, and she wasn't typing on the keyboard at all while all these windows were opening/closing on the website.
Also, from the bebop wikia page about Ein
In order to avoid having a human member of the Bebop crew be enlisted into the radical religious cult, S.C.R.A.T.C.H., Ein is put into the headgear that is used to link to the cult's website. Within seconds of being logged into their virtual lobby, Ein himself hacks into the underground portions of the site.