If you're playing on real hardware, you can connect to a compatible access point and change your DNS to the RiiConnect24 DNS.
You'll be able to use the in-game mystery gift download then.
If you're using an emulator, you'll most-likely not be able to use the DNS. I know that MelonDS supports WFC, but I'm not sure on other emulators. You'll likely have to manually import the Wonder Cards into your save file using PKHeX. You can find the Wonder Cards available for download at Project Pokemon.
""You should be happy. You're alive, and life is wonderful." - Ambertwo, PokÃ©mon The First Movie"
I was wondering if there is any way to use the claim mystery gifts button or if the special events that occurred would be available through the original rom or if they just have to be cheat-coded in. Any help would be appreciated, Thanks