Landry followed up his back-to-back 65s over the first two rounds with a four-under-par 67 at TPC Deere Run in Silvis, Illinois.
That was good enough to send him to the top of the leaderboard alongside Tringale, who carded seven birdies against one bogey on his way to a six-under-par 65.
Tringale is yet to win a solo title on the PGA Tour while Landry is chasing only his second career win following victory in the Valero Texas Open last year.
Tringale's low-scoring round came despite a bogey on the first hole. He recovered from that setback with birdies on the 2nd and 3rd, before making three more before the turn.
Two birdies and seven pars down the stretch brought him home at six under, for a 16 under aggregate.
"One of the things that's worked well for me this week has just been trying to play instinctively, not over-think, and go with my gut reaction," Tringale said.
"So even though tomorrow is a big day, a lot on the line, I'm going to try and default to instincts and not try too hard."
Landry and Tringale have little breathing space at the top of the leaderboard, with two players, 2011 FedEx Cup champion Bill Haas and Adam Schenk lurking just one shot back on 15 under.
Haas put himself in contention with a bogey-free seven-under-par 64 as he chases a first tournament win since 2015.
Haas launched his low-scoring round with a monster 43-foot birdie putt on the par-three third and never looked back.
Two more birdies on the 7th and 8th holes were followed by four more on the back nine to leave the American within sight of ending a miserable run of form that has included 16 missed cuts in 40 events since 2018.
"I've been working hard, probably working harder than I've ever worked at the game," Haas said. "I'm telling myself that those results will come. Hopefully they come tomorrow."
Just one adrift of Haas and Schenk are four players clustered at 14 under: Nick Watney, who also had a 64, Ryan Moore who carded a 65, Vaughn Taylor (66) and South Africa's Dylan Frattelli, who posted a 65.
But while the leaders were left dreaming of mounting a charge for victory on Sunday, it was a day to forget for Venezuela's three-time PGA Tour winner Vegas.
The overnight leader after a sensational 62 on Friday, which he described as his best round of the year, Vegas's hopes of a first win since victory at the 2017 Canadian Open went up in smoke with a disastrous five-over-par 76.