The Darkrai event in Pokémon BDSP as part of post-launch events, but Darkrai's event is now done, and the path to Newmoon Island has been shut for the time being.
From April 1-31, it was possible to acquire a Member Card in Pokémon Brilliant Diamond and Shining Pearl through a Mystery Gift event. Once the player has defeated the Elite Four and the Champion, they will earn the National Dex. With the National Dex in hand, the player can use the Member Card to enter the inn in Canalave City and sleep in one of its beds. This will allow the player to travel to Newmoon Island, where they can face Darkrai in battle, and attempt to catch it.
Darkrai was catchable in Pokémon BDSP, but the Member Card event has now closed, which is the same as the Shaymin event, as Oak's Letter stopped being distributed through the Mystery Gift option in March. The Azure Flute item for Arceus remains in the game, as unlocking it requires the player to have Pokémon Legends: Arceus save data with all missions completed, so that is permanently part of the game. The ability to visit Newmoon Island and capture Darkrai might not be active at the moment, but is there a chance that The Pokémon Company could bring it back in the future, in order to allow future trainers to fight the nightmare Pokémon?
It's Unlikely That The Darkrai Event Will Return
It's unlikely that the Darkrai event will return to Pokémon Brilliant Diamond and Shining Pearl. Usually, the Mythical Pokémon events only run once and then never happen again. This is part of a FOMO approach to the series, in order to encourage fans to buy the game as quickly as possible, as the window for legitimately acquiring these rare Pokémon is slim. It's possible that Darkrai could be directly given away on its own as part of a Mystery Gift event in a few years' time, but the actual Member Card event is likely done.
Even if the Member Card event doesn't return to Pokémon Brilliant Diamond and Shining Pearl, players will still have a chance to acquire Darkrai through other means. It's possible to trade Darkrai from another player and it's likely that there are a lot of them out there, considering that the event ran for a month. Once Pokémon Home, players will be able to trade one over from another game in the series. If Pokémon Legends: Arceus adds Pokémon Home in the future, then the easiest method will involve that game, as it's possible to unlock a Darkrai battle by having Pokémon Brilliant Diamond and Shining Pearl save data on the Nintendo Switch.