Rookie Vegas, the first Venezuelan to play on the PGA Tour, added a 67 to his opening 64 to lie 13 under par.
And he was joined at the top of the leaderboard by former Ryder Cup player Weekley, who added a 66 to his opening 65.
American duo Charles Howell and Chris Couch are a shot off the pace, with England's Brian Davis a shot further back on 11 under after a 66 on the Nicklaus Course at PGA West, one of four used for the 90-hole event.
Overnight leader Derek Lamely is five shots off the pace after struggling to a second-round 73, 10 shots worse than his opening effort.
But even that represented something of an achievement after he ran up a quintuple-bogey nine on his second hole of the day.
That came after a birdie on the first and was followed by four bogeys, an eagle and another birdie in a front nine of 41 featuring just one par, on the ninth.
To his credit, Lamely regained his composure sufficiently to play the back nine in 32 with four birdies.
Scotland's Martin Laird carded a second successive 68 to join Lamely on eight under par.
Vegas later revealed it was Tiger Woods who got him interested in pursuing a career in the game.
"I started watching golf on TV when Tiger Woods won the '97 Masters," Vegas told www.pgatour.com.
"That's kind of when I really started watching a little golf and developing that passion of wanting to be a professional golfer and so that's kind of one of the things that led me to be here."
Asked about leading a tournament for the first time on the tour, he added: "It's a lot of fun. That's the way I take it.
"I'm enjoying it as much as I can because we know how complicated golf can be."