In order to use the Friend Safari, you need to have traded/battled with somebody in Pokemon X, Y, Omega Ruby, or Alpha Sapphire at least twice. The first time, they need to accept you to be an acquaintance, and the second time, they need to accept you as a friend. This can be done online or with local wireless. I do not know if local wireless will break your DS, but if you can't use online, then that is the only way to do it.
Hope I could help