People who have watched or seen My Hero Academia are likely to ask the question "Is Overhaul now Quirkless?" because of the way the story plays out. After the events of his main arc in the series, Kai Chisaki is largely put on the back burner and doesn't have a major role to play in any events moving forward. This makes it easy to forget his state of affairs after everything he went through.
Once Overhaul was defeated by Izuku "Deku" Midoriya and the Pro Heroes, Tomura Shigaraki removed his arms, thus rendering Kai Chisaki Quirkless for the remainder of My Hero Academia. This explains why the character didn't play a role in the story afterward, as he was powerless to influence it in any way, especially considering the damage his sanity suffered because of what he went through in the series.
Disclaimer: This article contains spoilers from the manga/anime.
How Overhaul lost his Quirk in My Hero Academia

As most fans of the series know, Kai Chisaki, most commonly known as Overhaul, was the leader of the Shie Hassaikai and sought to take control of the criminal overlord to bring the yakuza back to the top. This was shown early in his main arc when he killed one of Tomura Shigaraki's allies, Magne, quickly establishing himself as a force to be reckoned with.
During this arc, he was attacked by the professional heroes and had to deal with the main duo of Mirio Togata and Izuku "Deku" Midoriya since they wanted to rescue Eri from him. However, after being defeated by the protagonist and taken by the authorities, Shigaraki and his men cornered him, and the villain removed Overhaul's arms, thus rendering him Quirkless.
Overhaul's localized Quirk Factor, explained

The question "Is Overhaul now Quirkless in My Hero Academia?" makes a lot of sense when considering that Tomura Shigaraki removed his arms by the end of the former's main arc. As mentioned earlier, yes, Overhaul, for all intents and purposes, is now a Quirkless man as of this writing in the series.
That is why he doesn't influence any event in the story in any meaningful way because he no longer has the power to contribute to the main conflict in any manner. His Quirk is localized in his arms, and without them, he can no longer execute his Overhaul ability due to the lack of connection to his hands.
Some fans have theorized that he could perhaps get prosthetics, as seen with the case of Mirko during the events of the final arc. However, since there is no biological connection, it would be impossible for him to get his powers back.
Final thoughts
It's understandable that people who have read or seen My Hero Academia ask the question, "Is Overhaul now Quirkless?" when considering that the story puts the character on the back burner after his main storyline. However, the series makes it quite clear that he doesn't have his Quirk anymore, and the prospect of prosthetics won't fix that.
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