You can register all Match Call entry after you beat the Frontier Brain Tag Battle on the Immortal Island. (talk to Flash when you finished the National Pokedex and got all Battle Frontier silver symbols).
You can rematch 62 Normal Trainers and Interviewer Couple Gabby &  Ty. For Wally and Sky, see Rivals Page. For E4, see Elite Four page.
Watch 2 videos below for the details: