I'm referring to his second lightsaber, which he built himself after Episode V, and then used in Return of the Jedi. Does he still have it?
They very deliberately showed Luke above the waist in the film. ALL through that scene. I'm not suire quite what they're trying to hide, but absence or presence of the lightsaber might be a possibility.
WGA Script doesn't say
Rey stares, knowing exactly who it is. But she just stares for what seems like forever. Until he finally TURNS, SLOWLY, to her. Pulls back his hood.
IT IS LUKE SKYWALKER.
Older now, white hair, bearded. He looks at Rey. A kindness in his eyes, but there's something tortured, too. He doesn't need to ask her who she is, or what she is doing here. His look says it all.
In response, Rey pulls something from the pack.
And she holds it out to him. An offer. A plea. The galaxy's only hope. HOLD ON LUKE SKYWALKER'S INCREDIBLE FACE, amazed and conflicted at what he sees, as our MUSIC BUILDS, the promise of an adventure, just beginning...
Foster novelization doesn't say (neither does Epilogue of Junior novelization by Kogge, which is worded differently but has same idea)
Whether motivated by her stare or by something unknown, the figure finally turned toward her and pulled back his hood.
His hair and beard were white, and his countenance was haunted. He did not speak, nor did she.
Remembering, Rey reached into her pack and removed his lightsaber. Taking several steps forward, she held it out to him. An offer. A plea. The galaxy’s only hope.
TL;DR: We don't know for sure.
They didn't tell us by design. Because Luke's Jedi robe seems to be open, which means had we looked at his waist we'd have seen it.
She OFFERS HIM the lightsaber in both texts. That seems to signify he doesn't have one of his own now but not sure. It could simply be symbolic; OR it could be a callback to Obi-Wan offering him that lightsaber in Episode IV to call him to the adventure.
Both texts very clearly call it his/Luke's lightsaber.
Not "hers" despite the fact that earlier the implication was that it destined for her.
This strongly suggests that Rey keeps the Anakin/Luke
hot potato blue lightsaber and uses it, which in turn strongly suggests that Luke still has his green one.
It's also worth pointing out that even if we did see all of Luke's outfit in the film, that wouldn't disprove the presence of the weapon because, post-Order-66, it's common for Jedi to conceal their lightsabers within their robes. Remember in Episode IV, how Obi-Wan had his the whole time and you never would have known until he whipped it out in the cantina? I suspect Luke is doing the same.
Spoilers for The Last Jedi:
The last time we see Luke with the green lightsaber, he is activating it in an effort to kill Ben Solo. He realizes the error of his ways immediately after activating it, but it is too late and Solo tries to kill him.
Using the canon book Bloodline, which is set six years before The Force Awakens, we know that
Ben Solo had not yet turned to the Dark Side. However, due to the combination of the revelation that Darth Vader was his grandfather (Bloodline) and that his uncle/master may have wanted to kill him, The Last Jedi, Ben turned to the Dark Side. So to answer the question, Luke had his lightsaber as close as six years before the events of The Force Awakens, and most likely lost it during the destruction of his new Jedi order.
Probably, but unconfirmed
In the novelization of The Last Jedi, the island caretakers on Ahch-To collect Luke's possessions after his death, including "his weapon":
Alcida-Auka verified that one of the daughters had cleaned the Outsider's robes and put them away in the storage hut, along with his woolens, pack, and boots. She directed another daughter to take his weapon, his star compass, and his strange other gear to the repository, where it would join other items gathered over the generations.
(Source: The Last Jedi: Expanded Edition, page 383 of Kindle edition, emphasis added)
Luke's green lightsaber isn't seen or mentioned at any other time outside of the flashback, but he isn't seen to use any other non-illusory weapon in the Sequel Trilogy, unless you count his fishing spear or an antenna he pulls off a hut while fighting Rey. Therefore, his lightsaber seems like the most obvious candidate for what this could be.
Note, however, that it's officially unconfirmed as of the 2019 edition of Ultimate Star Wars:
In the coming years, Luke continues to wield his blade as he rebuilds the Jedi Order. After sensing the dark side in his nephew, Ben Solo, Luke activates his saber above the young man while he sleeps, fully intent on killing him. Ben awakens and attacks his uncle, who immediately feels remorse. This moment leads to Ben wiping out the fledgling Jedi Order. After the Order's destruction, the whereabouts of Luke's second lightsaber are unknown.
(Source: Ultimate Star Wars (2019), page 282, entry: "Luke Skywalker's Green Lightsaber", emphasis added)