Tiger Woods, winner of three of his four starts this year, battled to an even-par 72.
A pre-dawn thunderstorm delayed the start of play by 30 minutes and another storm halted play for the day at 5 p.m. EST. The first round will resume Friday at 7:30 a.m.
Forecasts call for improving weather on Friday.
Kendall, Haas, Tway and Mediate all shot 68 over the blustery TPC-Sawgrass Stadium Course while Kevin Sutherland was at 4-under with two holes to play.
John Daly was in the group at 2-under 70 and defending champion Craig Perks was 2-under through 13.
Woods was tied for 36th place after his 72. He was 2-over through 15 holes, but birdied both the 16th and 17th.
"I wasn't hitting the ball as clean as I had been," said Woods, who last week won the Bay Hill Invitational for the fourth straight year. "I hit a couple of blocks, a couple of pulls, but I made some good putts to keep myself in it. Just trying to get to even par was a heck of an accomplishment considering the way I was hitting it today. I'm very happy to end (even par).
"You can never win a tournament the first day, but you definitely can lose it. I kept myself in the ballgame."
Woods reached the turn at 2-over par and the pivotal moment of his round came at the par-4 10th hole, where he made an eight-foot par save.
Mediate, meanwhile, did not have a bogey in his round of 68.
"I know the course, know where to hit it," said Mediate, whose home sits next to the second fairway. "I play 10 or 15 rounds a year here. I know what the swirls of the winds do most of the time. It's fun playing at home, but it can also we nerve-wracking at times."