There is a slot for SD-cards. If you'll be able to run unsigned binaries is still unknown. Games you purchase (through the virtual console stuff) will be signed so that they only run on the single Wii they were initially downloaded with.
(Actually, if you're interested, the serial interface between GameCube (and Wii) memcards and the system is very similar to the interface SD-cards use, so for example the SD-card adapter for GC is not much more than a socket. Memcards are a bit different as they have an additional "lock" circuit to make life difficult for 3rd party manufacturers.)