the DS don't have browser nor SD Card, so would need to store the homebrew on RAM, which he has a really small amount of. even with the opera RAM expansion.
the dsi (and the ds) can't access WPA2 encripted wifi, so....
to do this you will need to change your wifi to wpa (insecure), make a server that will send the homebrew over wifi (DS has no intenal memory) instead of some other site (DNS), and even then it will only send small HB 'cause of small ammount of memory on the DS...
too intensive for nothing.
on the other hand, go to the internet and buy a flashcard (r4) which costs 5$(!) and can load any rom easily, with just download-and-play interface w/o the need to do some silly exploits....
on the DSi that has already been done, but it can only run small homebrews. also too much effort.