There’s a trader in Ramanas Park who will give you these slates in exchange for Mysterious Fragments, though the only way to get those is by digging them up in the Underground, so hopefully you’ve been making use of that feature. Scattered around Ramanas Park are special stone pedestals with slots for slates that just so happen to resemble various Game Boy cartridges. This is a special location unlocked once you defeat the Elite Four and are entered into the Hall of Fame.