Even if it's possible than an SDHC card works, Nintendo probably doesn't want to make things confused, with newer consoles (=newer firmware) accepting SDHC and older ones rejecting them. Consider that not everybody has their Wii connect to the net.

It's still possible that they come up with a firmware that can handle SDHC cards and more than 2 GB, if older, smaller SD cards become real hard to find. At the moment, though, SD cards of pretty any size can be found in pretty any shop... well, unless you want a 16 MB card :)