If you send it to Nintendo, and they decide to replace the entire unit instead of repairing the one you sent, they should transfer all your data to the new machine for you.
There is no "online account" associated with you. It is associated with a specific wii console, so a different console would not allow you to access anything from the old console. The transfer of information (downloaded games, saves, shop channel information regarding previous purchases) would be performed by Nintendo, so that the shop channel will recognize your previous purchases on the old console with the new console. This goes for save game files for games with online components as well, since they are bound to a specific console.
If you decide to not send the console to Nintendo, and just decide to buy a new one, you would not be able to access your shop channel purchases with the new console.