There are often largish (van-type or station wagon) cabs waiting at Tokyo station. Can't rely on it, but it seems to be more common to see them there than at other places. They charge the same price as regular cabs.

The most pleasant, and likely fastest, way to get to Disney via public transit from the GH is to take the Hibiya line from Roppongi to Ebisu and then take the relatively new JR Rinkai line that runs about every 20 minutes from Ebisu to Shin-Kiba. There you can switch to the Keiyo line that goes to Maihama without having to walk a mile to transfer like you have to at Tokyo station. This is a bit more pricey as the Rinkai line adds about an extra Y250 per person for the segment up the middle of O-Daiba.