Already available on Nintendo's latest console, the Wii U, Amazon Instant Video is now procurable for the original Wii. The House of Mario announced today on the Twitters
that the streaming service is now downloadable through the Wii Shop Channel in the States. There's no mention of availability in any other region.
As noted in the handy-dandy picture above, the process is as simple as downloading the app through the system and then registering it by following this link we've got right here: Ta-da!