Reserving a "wagon taxi" in advance seems like a good idea but I think it's not practical for two reasons:

1. You can't predict when you will need the taxi since it depends on when your flight actually arrives, the length of the lines at immigration, etc etc etc.

2. It is almost certainly very challenging if you don't speak Japanese. I enjoyed the English web site you mentioned but couldn't even figure out the ENGLISH in the "reservation" portion -- which didn't mention the "wagon taxi."

Around 1/2 - 2/3 of the Hikari and Nozomi shinkansen stop at Shinagawa.