Johnson shot a 4-under 68, holing a 10-footer on 18 to match Landry at 13-under 203 at TPC San Antonio's AT&T Oaks. Landry birdied the 16th and 17th in a 67.
Johnson won the event in 2008 and 2009, the last two times it was played at LaCantera. The 42-year-old Iowan is trying to win for the first time since the 2015 British Open.
"I've got 18 holes to get to that point," Johnson said. "I've got to do exactly what I did on the back side and that was give myself opportunities on every hole. I'm putting great, I'm seeing the lines well, my caddie's reading the greens well, so it's just a matter of committing and executing down the stretch."
The 30-year-old Landry is winless on the tour.
"I'm a good putter and I just need to give myself a lot of opportunities tomorrow like I did today," Landry said. "I'll be looking forward to tomorrow."
Mullinax had a course-record 62. He played the back nine in 7-under 29, going 6 under on the last five with eagles on the par-5 14th and 18th and birdies on 16 and 17. He also birdied Nos. 10 and 12 and bogeyed 11.
"It's probably one of the best rounds I've ever had," Mullinax said. "To go out there and shoot 62 on a hard golf course is really good."
Johnson played the front nine in even par with two birdies and two bogeys. He birdied Nos. 11, 14, 15 and 18 on the back nine.
"Different wind today early on, misjudged some numbers, misjudged some wind, made some bad swings, all of the above," Johnson said. "But truthfully, my short game was actually pretty good, my putting was great. I missed some putts, but I hit some really good ones, hit some lines and I gave myself opportunities especially on the back side."
Landry had a bogey-free round.
"I just did everything really good," Landry said. "I was staying patient and just trying to make a bunch of pars. This golf course can come up and bite you in a heartbeat."
"I just feel like I'm getting closer and closer to playing better and better golf, more solid golf, putting rounds together," Walker said. "I'm excited for the opportunity tomorrow."
Mullinax has made 42 of 44 putts from inside 10 feet this week.
"They just kind of remind me of greens from home," Mullinax said. "My caddie, David (Flynn), has been reading them really well. We trusted each other on our reads and I've been hitting good putts. Been working hard on putting on the weeks off that I've had so it's good to see some results."
The 25-year-old former Alabama player chipped in for the eagle on 14 and the birdie on the par-3 16th.
"It was just a little bit down the hill," he said about the 16th. "All you had to do was just land it just past that little light grass spot. My caddie told me just read it like a putt, so I tried to just read it like a putt and it went in."
On 18, he hit a 3-iron from 255 yards to 15 feet to set up his eagle putt. He broke the course record of 63 set by Matt Every in 2012 and matched by Laird in 2013. The tournament record is 60 at LaCantera, by Bart Bryant in 2004 and Johnson in 2009.