Warning: The following contains SPOILERS for Legends of Tomorrow season 6, episode 8, "Stressed Western."
An episode of Legends of Tomorrow confirmed that the Justice League still exists in the Arrowverse, although they have yet to adopt that name officially. However, the episode did confirm that they formally adopted the Hall of Justice name for their meeting space. It also confirmed that, despite being one of the most earnestly heroic people in the Arrowverse, Nate "Steel" Heywood had apparently been passed over for membership.
The Arrowverse's Justice League was formed during the final episode of Crisis on Infinite Earths after Barry Allen revealed that he had constructed a round table with chairs bearing the symbols of all the heroes he had gathered to witness the lighting of an eternal flame at Green Arrow's memorial in an old STAR Labs facility. The same facility had been used as a base of operations during the Invasion! storyline and Barry hoped it might serve the same purpose again should the heroes of Earth-Prime ever need to join forces to save the world in the future. Unfortunately, the COVID-19 pandemic led to the cancellation of all the planned team-ups, leaving many fans to wonder if Barry's plan for a new super team had been abandoned, particularly in the wake of Black Lightning and Supergirl coming to an end in 2021.
In the season 6 episode "Stressed Western.", Legends of Tomorrow confirmed that the team still exists, in some form. While the Legends were in the Old West, joining forces with lawman and Diggle-doppelganger Bass Reeves against an alien sandworm who fed on anger, the ever-cheerful Steel finally lost all patience with his increasingly irritated teammates and went off on a lengthy tirade about everything that was bothering him that he had been keeping bottled up. The rant ended with him complaining about how he had "asked for a chair in the Hall of Justice" and never gotten a response.
Nate's venting was interesting for a number of reasons. The main one is that he indirectly confirmed that Barry's super-team still exists in some form and that there is apparently some process for heroes to apply for membership, even if it's just politely asking for a seat at the table. This begs the question of what the application process entails and who among the League members has a hand in making that decision. Whatever it is, it is apparently more involved than Nate pestering Sara Lance, who is the only member of the Legends to have a place in the Arrowverse Justice League.
The other interesting thing about Nate's rant in Legends of Tomorrow was that it marked the first time someone had referred to the meeting place of the Arrowverse's Justice League as the Hall of Justice in the modern day. Despite not knowing who is running things at the Hall of Justice, with Kate Kane, Sara Lance, and Supergirl all missing for most of the past season, there are clues fans can glean. Winn Schott, who traveled forward in time to join the Legion of Superheroes as the heroic Toyman, accidentally delivered a spoiler about the building's name in the Supergirl episode "Back From the Future - Part One." Presumably, Kara Zor-El codified the name shortly after that incident, but that still doesn't answer the question of who is running things at the Hall of Justice. There is still much to answer, but the show is on the way to doing just that.